Despite would work better... Despite can take the place of 'even though'; "Sally went to work even though she was ill." "Sally went to work despite the fact she was ill." In spite has a little more of a resistance factor. It really uses a more literal meaning of the verb to spite, in a slightly argumentative or malicious way... More like: "Sally went to the party in spite of her father telling her not to." Or "Sally came back late in spite of her boss yelling at her today." Kind of like when a bratty kid touches something you tell them not to. Though, if you use either (despite/ in spite) in any of your/my examples, no one will question the meaning.

They are very close. Apparently, despite is more formal http://www.englishpractice.com/grammar/spite/

Anha is correct but in everyday speaking I say and I hear the two used interchangeably