Learn Dutch: Moet ik het licht nu uitdoen? - Do I have to turn off the light now?

This sentence uses the modal verb moet (must/have to). When you use a modal verb, the main verb (uitdoen) gets pushed to the very end of the sentence. Because it is at the end as an infinitive (the full dictionary form), separable verbs stay glued together as one word.

Why does the sentence start with Moet ik instead of Ik moet?

This is a yes/no question. In Dutch, you form these by swapping the subject (ik) and the working verb (moet) so the verb comes first, just like saying 'Must I?' in English.

Doesn't doen mean 'to do'? Why does this sentence translate it as 'turn off'?

On its own, doen does mean 'to do'. But uitdoen is a separable verb (made of uit + doen) that is commonly used as a set phrase meaning 'to turn off' (for lights and appliances) or 'to take off' (for clothing).

Could I put nu at the very end, like in English?

No, because the sentence has a modal verb (moet), the main action verb (uitdoen) demands the absolute final spot in the sentence. Any other details, like the time word nu, have to go before it.