Learn Dutch: Zij spreekt Frans en Nederlands. - She speaks French and Dutch.

You would use welke. The Dutch word for language is de taal (a de-word). Because it is a de-word, you must use welke. You would ask: Welke talen spreekt zij? Remember, welk is only used for het-words.

Could I use ze instead of zij in this sentence?

Yes! Zij and ze both mean 'she' (as well as 'they'). Zij is the emphatic form, used when you want to stress that she is the one speaking these languages, perhaps in contrast to someone else. Ze is the unstressed form and is much more common in everyday speech.

Why does spreekt have a double 'e' when the infinitive verb is spreken?

This is due to Dutch spelling rules for long vowels. In the infinitive spreken (spre-ken), the first 'e' is at the end of an open syllable, which makes it a long vowel sound. When you form the stem (spreek) and add the 't' for 'she' (spreekt), the syllable is closed by consonants. To keep that same long vowel sound, you have to double the 'e'.

Are languages like Frans and Nederlands always capitalized in Dutch?