Advice for a Dad - 14 daughter tells me she's gay. 2 major questions.

I'm a widowed dad of 2 kids. My 14 year old daughter came out to me about 3 months ago expressing that she's gay.Question #1: I'm going to be honest here.. I'm supportive but I don't know if I believe her. I haven't let on to her that I question her sincerity but I do. Is that common for parents of teens? I'm I being completely shitty?Question #2: Sleepovers? My daughter used to have sleepovers every weekend. How can I allow her to have a sleepover with a girl in her bedroom? Even if they are "just friends".... Right?