"The X and (the) Y": When should I repeat the definite article?

In English, you are allowed to (but need not) drop repeated elements in parallel structures. The problem with your second example is that you are dropping the elements in the wrong order. The following are all grammatical, and mean the same thing (although the third is ambiguous as it could mean the request function or just the request):

• The arguments belong to the request function and to the data function.
• The arguments belong to the request function and the data function.
• The arguments belong to the request and the data function.
• The arguments belong to the request and data functions.

Your second suggestion,

• The arguments belong to the request and to the data function.

is correct if you mean to say that the arguments belong to the request, and not the request function.

Your first suggestion,

• *The arguments belong to the request and to data function.

is incorrect because you dropped the and didn't drop to.

For your first example, both sentences are correct.

This can be tricky, I'll start with your second example, since it has an unambiguous answer.

• The arguments belong to the request and to data function - This form is simply incorrect. In this case, the noun phrases "the request" and "(the) data function" belong to different prepositional phrases, so you should include the definite article in both of them.
• The arguments belong to the request and to the data function. - This is the correct way to state this.

Your first example is trickier, because in this example both noun phrases belong to the subject and are not split into two different prepositional phrases.

• The beginning and end of the channel - This is a fine thing to say, and is probably the most common way of stating this.
• The beginning and the end of the channel - This is also grammatical. Repeating "the" for a second time here gives a slight emphasis to the phrase "the end" and highlights the distinction between "the beginning" and "the end".

When in doubt, go with the first variant using only one "the".

Because you're using "and", by omitting the second "the", the meaning can be sligtly different.

Compare:

the hats and gloves

vs

the hats and the gloves

In the first instance, the hats and gloves are grouped as one and you're not referring to them individually.

This is not an issue if "or" is used.

Compare

the boys or the girls

vs

the boys or girls (this one reads wrong to my ears)