Я не помню, какое сегодня число.

Because число is a neuter singular noun, and какое is the neuter singular form of какой.

The forms are:

• какой — masculine
• какая — feminine
• какое — neuter
• какие — plural

So in какое сегодня число, the word какое agrees with число.

Also, in this expression, какое is best understood as what date, even though какой can also mean which or what kind of in other contexts.

It does literally mean number, but in this context число specifically means the day of the month, so the whole expression какое сегодня число? means what is today’s date?

For example:

• Сегодня пятое. — Today is the fifth.
• Сегодня двадцать третье апреля. — Today is April 23rd.

So this is a very common set phrase in Russian.

Because in Russian, the verb to be is usually left out in the present tense.

So where English says:

• What is today’s date?

Russian simply says:

• Какое сегодня число?

Literally, it is something like:

• What today date?

This is completely normal Russian grammar.

Because какое сегодня число is a subordinate clause attached to the main clause Я не помню.

Russian normally uses a comma before this kind of embedded question:

• Я не помню, какое сегодня число.

This is similar to English I don’t remember what today’s date is, but Russian punctuation is stricter here and requires the comma.

Russian word order is more flexible than English, but какое сегодня число? is the most standard and idiomatic way to ask for the date.

You may also hear:

• Какое число сегодня?

That is possible too, but какое сегодня число? is the very common fixed expression learners should know first.

So the sentence uses the most natural everyday order.

Because Russian does not ask for the date with что.

To ask what date is it?, Russian uses the pattern:

• Какое сегодня число?

Even though English uses what, Russian uses the agreeing form of какой with число.

So что сегодня число would be wrong.

They are close, but not identical.

• Я не помню = I don’t remember
• Я не знаю = I don’t know

In this sentence, Я не помню suggests that the speaker may have known the date, but cannot recall it now.

If you say:

• Я не знаю, какое сегодня число

that means you simply do not know the date.

Both are grammatical; they just express slightly different ideas.

Yes. It is an embedded question.

The direct question is:

• Какое сегодня число? — What is today’s date?

When it is placed after Я не помню, it becomes an indirect or embedded question:

• Я не помню, какое сегодня число.

So the structure is basically:

• I don’t remember + what today’s date is

It is in the nominative singular.

In the expression какое сегодня число, both какое and число are nominative neuter singular.

That is because the phrase works like a present-tense identifying sentence with the verb to be omitted.

So grammatically, it is like:

• Какое [есть] сегодня число?

with есть omitted.

Usually with a neuter ordinal number, because число is neuter:

• Сегодня первое.
• Сегодня пятое.
• Сегодня двадцать третье.

If you include the month, the month is usually in the genitive:

• Сегодня пятое мая.
• Сегодня двадцать третье апреля.

This is useful because it matches the question word какое.

It is understandable, but it is less natural in most situations.

• Я не помню, какое сегодня число. — natural, standard
• Я не помню сегодняшнее число. — possible, but sounds more like I don’t remember today’s date/number as a noun phrase

Russian often prefers the embedded-question structure here, just as English often prefers I don’t remember what today’s date is over I don’t remember today’s date in some contexts.

So the version in your sentence is the most idiomatic one to learn first.