I want to spray the trim with latex semi-gloss then come back and roll the walls with latex satin and roll the ceilings with a flat. Will this work? Will the sheens be affected by laying a lower sheen on a higher sheen?
Painting this way will work just fine and is a great way to save time and effort. This is similar to production painting, were the trim is painted first then protected with masking paper. Now the ceilings can be sprayed then the walls rolled. The walls could also be sprayed with careful use of cardboard spray shields.
I would at least spray the ceilings as these areas are the hardest to roll. The actual amount of spraying is depends on your experience and the amount of trim you would need to protect, but spraying as much as possible is faster than brushing and rolling.
There shouldn't be any issues with flashing or picture framing due to the layering of different sheens as long as the ceilings and walls are properly sealed and you use good paint.
I assume the trim is a new installation where spraying a bunch at a time will obviously be quicker. I would install the trim last in that case eliminating the need to tape which is also time consuming to do properly to prevent bleeding in spots when the tape is removed. I also lightly sand anything with a sheen for better adhesion. Spraying is great if you have a higher quality machine and most importantly experience with them. I also assume the room you are painting is unfurnished because although spraying is quicker, the overspray will leave a residue dust everywhere. I'd also start with the ceiling. Good Luck!