Comma in conditional sentence and in antithesis

Should I always put comma between condition and consequence parts, like in the following sentence?

If you have any questions, don't hesitate to ask.

You should. In the example you provided, "If you have any questions" is a dependent clause, and as such, it must be followed by a comma since it precedes the independent clause.

Should I always put comma in the sentences similar to the following (I may be wrong, but I think it's called antithesis)?

 It rains, but children continue to play football.

Yes, you should. In the example you gave, "children continue to play football" is an independent clause. Whenever "but" (or any other conjunction) is used to join two independent clauses, it should be preceded by a comma.

(Due to insistence)

If you speak out the first sentence, you'd pause in between "questions" and "don't", so a comma is definitely warranted. (If you have any questions, [pause] please don't hesitate to ask.)

For the second sentence, "It's raining, but..." sounds better to my ears at least, but your construction is not ungrammatical. Yes, the comma ought to be there, too.