Regarding games and letting her win: she's basically testing you to see whether you give in to her demands of letting her win, or you are more assertive/dominant and actually try to beat her.
If you pick the former, she knows you're a pushover and she can get away with far more down the line. If you pick the latter, she knows she's with a strong, assertive, dominant man who knows how to stand up for himself and does not get pushed around.
So long story short, it's not about the game itself, she's merely testing your character. It's a **** test and anyone who picks the option of letting her win automatically fails the test.