Should I be looking for a relatively stiff board if I don't plan on doing park?
You don't need a super stiff board unless you plan to be on powder. A good mid-flex board will work great all across the mountain.
I used the sizing guide/calculator from frostyrider, and it suggested I aim for a 156--158 (preferred 157) board. I'm 5'5.75" and currently weigh 150 lbs, although I think my weight will go up to 170 or 185-ish. It's difficult to predict how much I'll eventually add, so I'm trying to get some wiggle room for the added weight.
Going from 150 to 180 isn't going to give you much leeway on board length. That's a pretty big weight jump. If you really expect to get up to 180, then a 156-158 will work fine. (My groomer board is a 158 and I weigh 180). I would go smaller if you don't plan on putting on that much weight. For 150-160 lbs, you could probably do something in the low 150's. Board length isn't a science and a calculator is just giving you a rough idea. Your best bet is just to go with what feels right.
So I seem to have a wide selection, and not quite sure what exactly I should be looking for:
Selecting a specific board just amounts to doing the research and pulling the trigger. If you're really torn, do a demo and see how you like the board. Don't fret over it for too long, it's not worth it.
I am absolutely lost with boots and bindings. I know I should try both in shop and make sure they are compatible with each other. I'm guessing with a stiff board, I'll probably want a stiff boot too?
For bindings, you probably want a mid flex, but maybe something adjustable like Rome Targas. They are good all over the mountain. As for boots, I personally love BOAs, but it's a preference choice. If you're going to spend good money, spend it on the boots. They're the most important part of your boarding experience, they will make or break your day. Again, you probably want a mid flex boot. Too soft and you'll have issues in powder, too stiff and you'll have a lot less forgiveness on crud.