So is it really necessary to do this?
Yes. It looks like a bodge otherwise. If you lay without expansion gap, you probably will get cracks and bulges. You either have to leave the gap and apply beading, which looks rank, or you shave off the bottom of the door frame. It is very simple to do with thin bladed saw and a chisel.
I found it easiest to start on the wall away from the door and lay the door pieces near the end. Otherwise, it would be easy to faff around with the door stuff and not get a decent straight initial run, which you need to ensure you get few gaps and problems.
Christian.