What happens to male ejaculation after sex? Does it stay inside the woman?

Yes. If ejaculate entered vagina, it will stay there until it is expelled (it cannot get absorbed, as it is foreign to the woman's body and it would cause a rejection reaction), but the sperm will either die in the conditions of vagina or if an egg cell is ready to be fertilized, it may also do that, but it's only one, two, three... Sperm cells, but the rest will diet and get expelled.
Yes. If eyaculation occurred inside the vagina: yes. Some sperm cells may come out due to gravity. If the sperm cells find a fluid transparent and elastic they will survive for up to 7 days inside te vagina. If they (sperm cells)find fluid that is thick and non-elastic and opaque they won't be ble to survive. Female will perceive the seminal fluid for 24 hours. Bear down eliminates the fluid.