Does it matter if I mix neutrals and grounds in the main panel?
In the main panel, ground and neutral are connected, so you're free to attach to whichever bar is most convenient. There are two bars because if it was used as a subpanel ground and neutral would have to be kept separated.

The main breaker will cut power to the bus bars and breakers, but the lugs and feeder cables near the top of the panel will still be hot. Shouldn't be too hard to avoid those since the top slots are most likely full anyway and you'll be coming up from the bottom, just pay attention.

I'm not sure what code requires for routing, but it's probably best to drop the wire down in the neighboring cavity and cross through the stud at least 6-12" below the panel before coming back up to keep things nice and tidy. Or for a surface-mounted panel just route the cable neatly and anchor with cable staples: