Lots of good questions. For # 1, I would start putting her down for her naps without a diaper, and see what happens. From the way you describe things, it sounds like she is waiting to poop until she has a diaper on. You can buy thick "training underwear" for naptime, if she isn't already wearing them. Have her sit on the toilet right before naptime, and ask her if she needs to poop. Also, limit her liquids for 1 hour before naptime.
If she wakes up with poop, have her help clean it up. It doesn't have to be humiliating, just have her dump the poop in the toilet, rinse her underwear out, and change. No shouting, no yelling--just make it her job to deal with. When she realizes it's less work to just go on the potty, she will.
[To protect her mattress, buy a waterproof mattress cover. Top it off with a "pee pad", a fitted sheet, a second pee pad, and a second fitted sheet. Double-sheeting the bed allows for quick clean-up if she wets it. You can buy pee pads at pet stores--they are actually for housetraining dogs.]
She may still need a diaper for quite some time at bedtime. Some kids pee in their sleep and can not help it. Her naps w/out diapers will be a good test.
For # 2, I would just replace her potty chair. Sitting on the toilet will come in time; I wouldn't push that issue.
For # 3, set a timer (egg timer, alarm clock, or your cell phone) and when it goes off, it's potty time, and that means you, too, Mommy! Even if you just go in the bathroom, close the door, count to 10, flush the toilet and come back out, go potty before she does. Kids don't like to be singled out, and the peer pressure of everyone else in the house using the potty at set times can be a good thing. Also, with the timer technique, the timer is the one telling her to go potty--it takes the responsibility off of you.
Good luck.