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I recently learned about the notions of countable and uncountable noun in English language. I understand that "How many integers are there?" is a gramatically correct sentence. However, is the sentence "How many real numbers are there?" grammatically incorrect? I think it should be since there are uncountably many real numbers. Also, if we assume that the continuum hypothesis is false, then what is the appropriate determiner for the intermediate cardinal? Is it "many", "much", or another word?

You are confusing grammar and semantics. You are also confusing the idea of a question having an unusual answer with it being an invalid question.

Just because the answer to the question "How many real numbers are there?" is "An uncountably infinite number" doesn't mean it's an invalid question, still less that it is ungrammatical.

A mathematician would say that the question is a useful one, as he might want to distinguish between the different types of infinity.

Other questions that are perfectly grammatical, if lacking a useful answer, might include "What colour is the wind?" and "How do your colourless green ideas sleep?" (The answer being "Furiously.")

You may also be confusing the idea of an "uncountable" noun with an "uncountable infinity". An uncountable noun is one where you cannot enumerate the noun at all. In mathematical terms it's more akin to the difference between "continuous" and "discrete". I cannot count "water" at all, so I say "How much water?". I can count some real numbers. For example I can say, "I have written two real numbers down". So I say "How many real numbers"

There are three issues here.

The first is that you cannot base the grammar of the question on the answer which you do not yet know. You cannot say

It should be "How many are there?" if the answer is plural and "How many is there?" if the answer is singular.

That is just not possible. Each language has to have a convention about the grammatical number of How many that does not depend on the answer. In English we treat it as plural. In Scots Gaelic it is treated as singular. But either way it is just a convention, since we have to use some grammatical number before we know the answer.

The second issue is that countable means different things in grammar and mathematics. In mathematics, countable means you can put them in order to at least start the process of counting them all. Whereas in grammar, countable means that you can count even some of them. If you can say you have two hands, one head and no tail then you have counted some of the appendages that exist in the world, without even considering if it possible to count them all.

But the third, and most fundamental point is that even our modern grammar is based on an understanding of the world that predates any human concept of any number other than the so-called natural numbers (1, 2, 3 etc.) and was beginning to take shape when we were beginning to contrast "one" and "more than one". (Think about it - how could you tell someone there was more than one of something before the necessary language had been invented.) This means that it is not a reasonable assumption that existing grammar rules cover non-natural numbers. This is why people still argue about questions like "Is 0 singular or plural?", "Is -2 singular or plural? "Is 1/2 singular or plural?, "Is 1+i singular or plural?". These are questions that cannot be answered by logic or historical use because these numbers were invented after grammatical number.