Hôm qua, An đã ngủ rất muộn vì xem phim.

Hôm qua means “yesterday” and sets the time context. Vietnamese is flexible with adverbials, so you could say
– An đã ngủ rất muộn hôm qua vì xem phim.
but placing Hôm qua at the front (time–topic position) makes the sentence more topical and natural when you want to emphasize “yesterday” first.

Not strictly. Hôm qua already tells you the action is in the past, so you could omit đã:
– Hôm qua, An ngủ rất muộn vì xem phim.
Including both is common for added emphasis on the past aspect or simply out of habit, but it’s not ungrammatical without đã.

rất muộn literally means “very late.” Seeing rất before an adjective or adverb intensifies it. You could also say muộn lắm (“late very”), which is equally correct but a bit more colloquial. The nuance:
– rất muộn feels slightly more formal/standard
– muộn lắm is more conversational

1. vì introduces a cause/reason clause and does not require its clause to carry tense particles.
2. In Vietnamese, once the subject is clear (An), you often drop it in the subordinate clause. So vì xem phim is concise and natural. You could say vì An xem phim, but that’s more explicit and sometimes redundant.

– bởi vì also means “because” and is interchangeable with vì, though bởi vì can sound a bit more formal or emphatic.
– vì vậy means “therefore” (result), not “because.”
– nên means “so” or “thus” (draws a conclusion). You could rephrase as:
An đã xem phim đến khuya, nên hôm qua ngủ rất muộn.
but that inverts cause and effect and uses nên to link them.