They remove the placenta manually - meaning that they put their hand between the uterine wall and the placenta to shear it off.
The amout of bleeding after the birth is typically the same. There might be more clots for a cesarean mom if she isn't moving around and walking alot (the clots are just blood that pools for a length of time, then comes out when you stand or move a certain way).
It varies so much from woman to woman. I have many clients who are still spotting or lightly bleeding at 4 weeks, while some are completely done at 2. Then others are still having small bits of blood even at 6 weeks.