Amy girl
YEs your placenta is removed through the incision in your belly.
THe anaesthetist usually gives you synto through your IV - you can discuss this with the anaesthest at your pre-anaethetic meeting with them.
They will need to close you up as soon as they can to avoid infection.
There will still be a little blood built up in your uterus after your belly is sewed up but they will apply gentle pressure to your abdomen to push this out vaginally, it doesnt hurt as you are numb.

hth