What hardware if any should be replaced every time the brake line comes off the caliper? I know there's a brass washer for the banjo bolt, but what is the criteria for replacing it?

New/re-furbished calipers should include new washers. Always replace them when changing a caliper or line.

    I'm going to do pads on both sides of the front since they're getting close anyway. Am I correct that there is no need to replace the other caliper as well?

Yes, you do not need to replace the other caliper. You should inspect and clean it as much as possible (a toothbrush works well).

    Can I treat the threads on the bleeder nipple with anti-seize or is this just asking for trouble from the anti-seize contaminating the brake fluid?

No need to because you only tighten these very lightly. You just tighten them until they are snug.

    What generally is the torque spec for the bleeder nipples? The Chilton / Haynes books are mute on the subject.

Just enough. Trial and error works here. As a general rule:

 - Screw it in with your fingers. 
 - Use a wrench and go until you feel a bit of resistance.
 - Give it a light nudge without moving more than a couple of millimeters.

Pump the brakes. If they don't leak, then you are done.

    When the bleeder nipple came apart, I had my son press the brakes as far as they would go, and they are firm, and I watched the bleeder and no fluid escaped. Am I correct to conclude that this is safe to drive until I can get the caliper on there?

It is only safe if you are absolutely sure it is not leaking fluid or letting air in.