Can I split up time phrases and adverbs? If there is more than one part to a time phrase (e.g. both 今天 and 八点) can I place one of them before the subject and one after? Is there any situation in which this would be okay?

他今天八点要去看电影 (not split)
vs.
今天他八点要去看电影 (split)

On a similar note, if there is both a time expression and another adverb, can I split those up?

电影六点就开始了 (not split)
vs.
六点电影就开始了 (split)
The standard word order is Subject Adverb Verb Object.  Time adverbs are much more fluid in positioning than place adverbs, i.e., can do S time-adverb V O, or time-adv S V O.

The difference is what I call "focus" (I distinguish between "focus" and "emphasis", as there are emphasis words, e.g., 并 to go with 不, emphasizing the negative).

他今天八点要去看电影: sentence starts with 他, so the focus is on 他 (not 我, not 你), i.e., What's he going to do?

今天他八点要去看电影: sentence starts with 今天, so the focus is on 今天 (not yesterday, not tomorrow), i.e., "What's happening today? He is going to see a film at 8 o'clock" (whereas I'm going to do something else, say; implied, not necessarily overt contrast).

Ditto with:
电影六点就开始了: sentence starts with 电影, so focus is on 电影.
六点电影就开始了: sentence starts with 六点, so focus is on 六点.

It's what I call (in my teaching) the linguistic flagging up equivalent of physically holding up 1 of, say, 3 objects.  All eyes will look at the object held up, so when you say, "It is...", they know which of the 3 objects "it" refers to.  So when you start with 今天, they'll focus on 今天, not 昨天 or 明天.  When you start with 他, they'll focus on 他, not 我, not 你.