Not certain but I don't think OB's want to use C-section time to sterilize women. There's just too much going on. They want to get the baby out, safely, & stitch you up before possible complications set in. Later, after you recover, they can tie your tubes. (Your fallopian tubes are located well above the C-section area. To reach them they make a tiny incision near your belly button & insert a narrow surgical instrument.) However, you can always ask, right? Maybe the doctor might be willing to do it, with proper planning?
No, you can have it with natural birth. With a C-section they will do it at time of delivery while your still in the OR. With natural birth it could take a few days to schedule it and then they will have to make a small incision by the belly button to do it.