As far as getting the call based on if you live near a base;
I would say that if you DON'T live near a gateway and you are willing to relocate to a gateway then you need to make that VERY obvious on your resume.
It's a bit of a non-starter if you live in bfe and can't report for duty. With that said we just got a commuter policy that allows crashpads, relatives and the sort...so it's possible I guess. I'm not very familiar with how it works as when I was hired we had to live in our gateway (2 hour drive anyway).
As for changing your base, well...if you live at one gateway and you want to move to another one that's open then you simply call the chief pilot and get the ball rolling. If it's open then you get it based on seniority.
All schedules are 7 on and 7 off with every day being a start day. Weekend start days go junior and tues / weds go senior. You can bid a start day as often as you like but there is some pain involved to get to a new start day (read extra days at the daily rate) so it's best to just get to your preferred day and stay there.
Fly Safe.