Is there a downside to grouting part of a room?

If you're talking about the outer edge row or two and "later date" is a weekend or two from now, I see no real problem.

But, unless I had some other overriding reason, I'd probably just wait and do the whole thing at once rather than go through the mess and cleanup a second time...

Depends on a bunch of factors. Color of grout, how accurate you are at measuring the grout mixture and limiting water during cleanup, etc., but in general it's not a big problem with lighter tone grouts and a delay of a couple or three days. That being said, I'd probably wait and do it all at once. It's not like you can really use a partially done shower or bathroom and it's easier to keep going than to stop and start again.

I agree with the others. We experienced grout color differences when we did grout in smaller batches. Each batch was slightly different. You could run into the same issue if its days apart. Less messy if you wait and do it all at once.

ScottI still have to cut and lay the tiles around the edges. I'm probably

Are you supposed to mix bags of grout to combine the color? I think I have read that .. don't mix a whole bag in one batch, but save part of the bag to mix with the next?