Learn Dutch: Zij stelde veel vragen. - She asked many questions.

In English, you 'ask a question,' but in Dutch, you cannot say een vraag vragen. You have to 'pose' or 'put' a question, which uses the verb stellen (een vraag stellen). Stelde is the simple past tense of stellen for a singular subject.

Is vragen a verb or a noun in this sentence?

In this sentence, vragen is a plural noun meaning 'questions' (singular: de vraag). It looks exactly like the verb vragen (to ask), but here it functions as the object she is asking.

Why doesn't veel get an -e at the end (vele)?

When veel (many/a lot of) is used before a noun to express a general amount, it usually doesn't take an -e ending. Vele does exist, but it sounds very formal or poetic. For everyday Dutch, veel vragen is the standard.

Could I use ze instead of zij here?

Yes, ze and zij both mean 'she' in this context. Zij is the stressed form, used when you want to put emphasis on the person ('She' asked many questions, not someone else). Ze is the unstressed, everyday form you will hear most of the time.