What age? If the pitcher is one of the best athletes on the field (and the 1st baseman is not) at 8u, 10u or 12u then let her take the cuts. At 14u we gotta get better. IMHO. 1st base is an important position and often underrated.
Yes we use the pitcher as the cutoff on throws home from the OF. It allows us to "cut" and possibly have a play on a runner if the IF's cover properly.
Certainly can. Lower level teams your pitcher may be a better athlete than your first baseman, but as you move up in level and get to teams where you have some PO type pitchers you probably want to be using your first baseman.