Learn Italian: Il modulo chiede il nome e il sesso. - The form asks for the name and the sex.

No, this is just the regular present tense meaning it asks. If you wanted to give a formal command to a person telling them to ask something, you would use chieda, because formal commands for -ere verbs take an -a ending. Here, the subject is il modulo (the form), so it is simply a statement of fact.

Why doesn't it say chiede per since the English translation is asks for?

In Italian, the verb chiedere means to ask for all by itself when it is followed directly by a noun. You do not need to add a preposition like per.

Is it normal to use sesso on an Italian form? What about genere?

Yes, sesso is the standard word used on official forms and documents in Italy to ask for legal sex or gender. You might also see genere, but sesso is still the most common term in bureaucratic contexts.

Does il nome mean the full name or just the first name?

Strictly speaking, il nome refers to the first name, while the last name is il cognome. However, if a form only has a single box labeled nome, it usually expects your full name.