Is this connection casual for her?

My girlfriend recently got asked out on a date by someone she is friends with on Facebook, and she just laughed the incident off. When I asked her if she is considering meeting him, she said she wasn’t going to, but she also said it was harmless and not reason enough to stop chatting with him. This guy knows she is in a relationship with me because it is clearly mentioned on Facebook, but that didn’t stop him from messaging her. If this doesn’t bother her, am I to assume she doesn’t take me or our relationship seriously?
Why assume when you can ask her about it and give her a chance to explain why she looks at it differently? Yes, the guy who asked her out was crossing a line, but people do that all the time, and she may laugh it off because she isn’t taking him seriously. To assume this is a sign that her relationship with you is casual seems misplaced. You can’t control who she can or cannot speak to, but a solid relationship is built on trust. When you manage to do that, these interactions will cease to be important to you too.

I have been dating a guy whose father doesn’t like me at all. I have tried everything to get along with him, but nothing works, and I think it’s because of my background because my family is not as well-off as his. If he doesn’t change his attitude towards me, I worry that my boyfriend will break up just to avoid the pressure. What can I do to win his father over?
You can’t always win someone over if they have fixed ideas about who their children should or shouldn’t be with. It makes more sense to tell your boyfriend about your fears, and trust that he has a mind of his own that allows him to make his own decisions. Keep meeting his father if possible though, because time also gives people a chance to change their minds.