Ok, a couple things:

1) Yes she is sounds a lil insecure. really though you are the one to judge her. But I'll say this, it depends on what you want from the relationship ie do you just want something casual or are you looking for a real girlfriend. Maybe what you want and what she wants are different.
2) really if you are not hiding anything, it should be fun to watch her go through your stuff. I make a joke about it all the time with my lady.