If you replace the caliper, I'm an old-school firm believer that you should bleed all 4 corners to get all the air and old fluid out. No sense in having a water/rust filled fluid pumped into your fresh, new caliper. Some people seem to think you can just clamp the hose, swap the caliper, then open the bleeder and let it gravity bleed until fluid comes out. I'm not one of those people.
Personally, if you have a stuck caliper that got so hot that it turned the pads red, you should 100% replace the rotor. It'll be so heat-warped that you've have a persistent vibration (not a huge deal on the back) from the out-of-round rotor. If you really want, you can just replace the rotor on the stuck side (along with pads on both sides) to save yourself the $100(ish).