Re: 3 year old won't "poop" in the toilet
With our daughter we had to make her naked from the bottom down. We had the little potty out and she just went because there was nowhere else for her to "put" the poop. I guess there's a risk of her just going on the floor. But my daughter would poop in her underwear. So if I knew she needed to go, I would just take everything off and ignore her. Asking her to go made it much worse. She would have a fit, so we decided to ignore her and not say anything at all about going on the potty. Finally she got uncomfortable and would just go and sit on the little potty and go. We made a huge deal about it and eventually she just did it. Now my daughter was 2.5at that time, so a 3 year old might be a little smarter for it, but it's worth a try.
For what it's worth, my oldest daughter was like yours. She was 2 and would pee, but didn't poop until 3. I tried everything, rewards, wrapped gifts, bribes....she eventually just did it.