Yes... and no...

Is today's communication fast enough and capable enough to manage a planetary empire? Absolutely. The Internet alone could do it.

That's not your problem. Let me suggest a frame challenge:

Your problem is establishing such an empire in a believable way, because the technology that allows massive, instantaneous communication also allows for considerable freedoms involving transportation, weaponry, housing, and (not surprisingly) communication against the empire.

When you consider the various near-worldwide colonial/empire periods of Earth, you discover that people were (relatively) easily controlled because the conqueror had tremendous advantage in firepower and manpower. Genghis Khan was attacking, predominantly, villages that were loosely organized at best. England, Spain, and the Netherlands brought overwhelming firepower and organization to bear against the many lands they conquered and controlled. The same can be said about the U.S. westward expansion.

You don't have that advantage today.

It seems like you do. You have tanks and planes and aircraft carriers — and they're all very impressive — but while you may logistically control the entire planet, you would be constantly fighting rebellions and resistances. Well-armed rebellions and resistances. And history has proven that they could fight against tanks, etc. in the past.

Technology is a great equalizer (well... for bad, if not for good). Consider the various difficulties in Africa today — difficulties made simple because arming yourself with automatic weaponry that can eliminate whole villages in minutes is trivial. Why can't the villages do the same? They're poor.

Which is probably the only way a planetary empire could stay in control — poverty. But then, what's the point of the empire?

Yes.

Many large corporations and organizations already operate on a global scale just fine. Just add some local and regional governments such as nearly all governments already do and you have a functioning global empire.

It is unlikely to happen with a democratic government without an alien invasion or some other reason to have world government but a fascist or communist government might go for world domination. Their legitimacy comes from a unifying ideology and works just as well (or badly) for a planet as it does for a single commune.

A theocratic government would also work. An Islamic caliphate for example would actually have perfectly valid reasons to seek to unite the world under its power. A Christian or Hindu version of the same would work just fine.

And manufacturing a reason for a democratic government to go for it isn't really that hard. Any kind of global event that threatens the extinction or subjugation of entire human species would work. Or people could just get really tired of pointless wars. The European Union largely exists because people figured that an economic union between France and Germany really would save lots of bother.

EDIT:
Since I started by pointing out that corporations work on a global scale, I probably should mention that it has required some adaptations to how they operate. This is because the "working hours" happen at different times in different parts of the world. Typically (a simplified general model) you'd split the world into three operational zones (24/8 = 3) and have active operational control cycle between these. Typically these are the Americas, Europe and Africa (or EMEA), and Asia and Oceania.

A global empire would probably copy this solution and essentially have three administrative centers. Say, Washington, Brussels, and Beijing. Depends on how the empire came to be. The central government could be in one of these or cycle between them or even be divided between the three. The EU has split its functions between multiple locations for political reasons and it kind of works with fast travel and telecommunications.

Yes. I'll break it into two parts, communication relating to Empire Citizens, and Communications within the powers that be.

My first thoughts go to China and their citizen points system being worked into place. The first step would be to pull as many people into the cities as possible, having most of your population in sight of you cameras and tracking technology would be crucial. Monitoring any internet/ media usage of a population is a common occurrence today, and would be extended to make sure nobody is forming a group to go against your reign. We have the technology to track the whereabouts of every citizen, so we would be able to tell who is meeting up with who, where people are going and what they are doing. With no privacy, people cannot create a revolution.

Communication between generals, media controllers, the world leader, and everyone else is instant. People are able to make a snap decision and have millions of people respond and react to it, I don't see why that couldn't be scaled up to billions. For example if a curfew was implemented suddenly, the police/ military could immediately respond to the situation without having to take the step of waiting for a courier.

Most of these solutions are dystopic, but I think world government and control over the whole planet causes those situations to arise. You would need to squash all attempts at disruption, and you would be able to do so quickly.

This is the bare minimum needed for an empire to control the whole planet: Optical telegraph, trains, mirror-based telescope, masts, airplane wings, fans (of any kind) and hot air balloons (to mix them to get an airship), and reduction gears. Extra points for steam engines, stirling engines and windmills, waterwheels and aerial ropeways.

• The optical telegraphs are used to send and receive messages.


• Mirror-based telescopes are used to check on other nearby settlements, to see if -something is wrong, and for mapping.


• Airships powered by pedals or treadles for mapping and traveling without having to use roads, or to change from carts to ships and back. Reduction gears would be used for moving the fans at the wanted speed.
  
  
  • Trains powered by pedals/treadles/hamster_wheels and using human or animal power to move, and aerial ropeways to help it move uphill or accelerate faster when leaving the station. Reduction gears would be used for moving the wheels at the wanted speed.
  
  


• Steam engines for ships and airships (not necessary airplanes), for when the masts are not delivering enough power, or when you would rather hurry.


• Stirling engines are for mechanical power generation, and for refrigeration if you run them in reverse, feeding them movement and moving heat instead of feeding them temperature difference and receiving movement.

Using tall towers for the optical telegraphs, you can increase the distance you can talk with both fixed and moving targets. Usually, the fixed targets will be talked with by using strong lights and mirrors to blink/flash towards the target, and the target will use a telescope to look at you and see the signal. For that reason, it is likely that different towers or different coordinates will be used, often with some space nearby to make sure the target is not thinking you sent a message when in fact it was a different tower.

The optical telegraphs would be first used for mapping, and next would be using airships for mapping. Due to the danger of airborne attacks, the medieval castles will evolve to oversized ziggurat cities, with mirrors sending a lot of sunlight to the underground, since using torches would be rather dangerous.

With airships, you can get a better picture of the continents, also useful for maps, and also travel for long distances without worrying about mountains and water currents, but instead worrying about wind currents.

We had bearings, both plain and roller and ball bearings since around the time Roma was born. There were also trains used for a short time period around that time, too. Even without steam and stirling engines, just with wind power, water power and pedaling, we can have an empire span the whole globe.

The trains would be used for moving things on a long distance using windmills, waterwheels and pedals/treadles/hamster_wheels.

Stirling engines can be used instead of steam engines and watermills and windmills for a distributed mechanical power generation network, probably using mechanical tension on ropes to move the power around. Alternative ways to move power would be compressed air, compressed water (tap water) from a water tower (or more water towers) used as storage and compression chambers. The energy can be supplemented with wind power and/or water power, if needed or wanted.

That's it. That's all one would need. And that is still a bit much.

Edit: You can use pedal/treadle powered vehicles to move faster with either human power or animal power. Just get an animal or human on an oversized hamster wheel (like those thread wheels used in medieval times to move heavy things up in elevation), and you can use their output to achieve high speeds. With or without rails. During the US's Great Depression, people tried using pedal-powered cars, but shortly after that there was a lowering in fuel price, which caused the idea to be abandoned by most people.

There is a lot more which can be done with that, so ask me if you want to know more.

Edit 2: It would also be useful to have a technological advantage over the rest of the world, to be able to give important gifts to the taken over people, to win them over and ensure rebellions will not happen. Since papyrus can be made, and wax and resin were used to waterproof things, there can be an empire which uses waterproofed papyrus paper boards for greenhouses, which might be cheaper and require lower tech than glass does, plus that you would use glass for the mirrors needed for the telescopes for the optical telegraph. Greenhouses will also keep your existing people fed and give them cheaper condiments. For growing temperate plants in the desert, you would likely need some way to cool the air, like through pipes going underground for tens of meters, before getting into the greenhouses where the amount of light is limited by having a few small windows allowing a lot less sunlight inside, which paired with the colder air can sustain temperate-climate plants. The humidity would still be an issue, though. Being able to get spices and grow temperate plants in colder and hotter climates surely does make up for that.

Yes, absolutely. It will be easy to control the entire world, once dominance is established. And rebels will not be as strong as they were, in the past. Yes, rebels do use modern technology at combat, but because the tech today is massively destructive, and the army uses modern means of control, science and logic in reasoning and deducing strategies, it is quite unlikely that any rebel team would succeed at outsmarting the empire. They would be better off working from within the empire to betray the emperor, but that will also be quite risky.

Why is that? In the past, people never dynamically employed scientific and logical strategies in combat, as much as we do today. Most kingdoms and empires in the past were taken down by outsmarting the emperor's army. But today, the military has R&D, and thus creates advanced weapons. They have heavily funded teams of brilliant minds who would work towards taking down their enemy. Today, the rebels do not have enough space to hide, as every part of the earth is measured and mapped. Even if they hide, it will not be possible for them to grow big and become dangerous: because they will need a lot of R&D and training, thus must have a well funded team of scientists, strategists and other people skilled in combat.

If someone had already managed to become an emperor (which is highly unlikely) it will literally be impossible to take that person down. Because, he will have the best minds on the planet (literally on the planet, as he controls the entire world), and as long as they work for him, and as long as he is in good terms with people who are immediately below his rank, nothing will stop him.

There is a new emerging psychology called the cyber-psychology. The same methods used by the game industry and the social networking sites can be used to massively control people. It is possible to psychologically control people by socially reward them for certain task while punishing them for the others. Imagine using a social networking site in a imperial world. If the emperor hates certain kind of topics, it is unimaginable to think that he wouldn't socially punish citizens who engaging in such topics. A simple way to do this is by improving the visibility of their content (which involves their personal profile or anything else they post) among people who would genuinely despise the topic and decrease the visibility of the content among people who may have the same views. Actions like hearts, likes, up-votes or points etc can be used as the incentive here. This could also go further- to jobs and credit scores. Soon, most content in the social network will be in favor of the emperor. And massive surveillance can be used to study people and respond almost spontaneously, to "fix issues". In this situation, there is no room for rebels. For rebels to exist, you need a mob of angry people, not zombies controlled by social networks. Also, even if you have angry people, what chance will they have against heavily funded experts working for the emperor?

Thus, I believe today is the right time for someone to be the world's emperor, as he could do more than just "communicate very fast". He could literally condition the psychology of the world to his needs. But it is highly unlikely that someone would grow powerful enough to conquer the world.

Yes

The limitation you provide has been overcome for centuries. The British Empire was globe-spanning, and could keep itself together, project force, deal with insurrections, etc.

It had sigificant ability to project force inland -- British India is not a coastal nation. It even fought wars in Afghanistan, one of the least navaly-accessible regions of the world.

More recently, both the American and Russian empires where globe spanning. Both ruled through proxy or puppet states, with the Russians being a bit more heavy handed.

The Russian empire crossed Asia, and included states in the Middle East and Cuba. It even fought wars in SE Asia; the logistics of having a semi-hostile China between it and the region, and American naval dominance, was more of a problem than communication delay.

The American empire included naval and military bases scattered over the world (which still exist), and "allied" puppet democracies, dictatorships, and kingdoms. They had allied empires (the old European ones), and could project force around the world in response to insurrection (such as Vietnam, an insurrection against the old French empire).

The 3 large Empires left -- America, the fragments of the cold war Russian, and the Chinese -- have different approaches.

The Russian Empire is at this point honestly too weak to project force much outside its borders.

The Chinese and American are mostly trade-based empires. Chinese conquest of Tibet/Mongolia and the South China Sea, and its threatened conquest of Taiwan, are relatively local geopolitical concerns; it lacks the ability to reliably project force past an American naval screen. It may or may not become more openly imperial as its economic and military strength grows.

American's traditional approach has been to use trade to expand its imperial influence. So long as you allow US companies in to reap profits, it leaves you alone. If you do go protectionist and you are relatively small and you align with regional or global foes and there is something the US could profit from in your territory, you are basically placed on a "oh well" list, and the next time an American president needs a poll boost they start talking about an "axis of evil" or whatever excuse and replace your government.