There is no way sperm can stay alive for 10 days! Thats insane.The longest ever was 7 days but thats only if your body holds the proper conditions.
Answer
sperm exposed to room air on clothing, bed linens or toilet seats lose motility (the ability to swim) rapidly. Once the semen dries out, the sperm are usually dead.
That said, you should take basic caution after sex that you've cleaned up any big splashes on her or around her.
Answer
if the semen is near the woman and no fertile fluid is present, they will die within a few hours in or out of the body. If there is fertile fluid from her, they can live for a week waiting for the egg to be released (which probably won't wait a week to be released with fertile fluid present).