In the sentance, " How many nights do

[hasty moon]

Why can't it be just "Combien de nuits vous voulez rester? "

[analog pivot]

You can but that's informal. In questions, you either have to invert the subject and verb (formal), add est-ce que (neutral) or leave it as is and let intonation do the trick (informal)

though with informal it's more likely the question comes at the end

[Formal] Inversion: « Combien de nuits voulez-vous rester ? »
[Neutral] est-ce que : « Combien de nuits est-ce que vous voulez rester ? »
[Informal] Intonation : « Vous voulez rester combien de nuits ? »

jsyk we do the first one in English, it's just that it's somewhat 'invisible'