A topic I actually have some expertise in!
I remember getting my GF into FPS games. Just a couple years ago she was playing BO2 and getting murdered by bots on hijacked. She was as bad as you can get. She would take five seconds to line up a shot, and even then she was likely to miss. About 18 months later she was making callouts, rotating on Dom matches, and had about a 1.8 KD and over a 2.5 W/L.
We have played several hundred hours of games together. In that time, here is what I have learned:
(1) Dudes are going to chat up your girlfriend. Most will be polite about it, and occasionally they will be graphic and vulgar. It is going to happen, and there isn't anything you can do about it.
(2) Your girlfriend is perfect capable of dealing with these advances. Most likely, she spent hundreds of hour dealing with this type of shit offline before she met you. She is the expert on this, and you do not know shit. She can handle it. She has a black belt in thwarting mens advances. You don't need to defend her on this one.
(3) It is 100% not weird to have some of your close gaming friends be Facebook friends. I have a half dozen of them added as does she (mostly the same people, but not all). After you spend 100 hours talking with someone, and you know everything about them, being Facebook friends is very reasonable.
(4) For the love of God buy your girlfriend her own console and play with her!
Anyway, to the question at hand....are you acting like an insecure douche?
The painful answer, brother, is that you are. And the fact is, it sounds like it predates you two playing video games. Every couple that has problems with Facebook and decides mutually to delete it (or worse create a joint account) is not a couple with a solid foundation. One or both of you is extraordinarily insecure.
Maybe it's justified. Maybe it's not. But either way it is completely unhealthy.
Look at it this way: you are worried about what she does when she is online playing video games? What about when you are at work? Or when she is at work?
YOU CANNOT CONTROL YOUR GIRLFRIEND!
You have to trust her. And if you can't, then you need to find someone else. And if you can't find someone you can trust, you need counseling.
TL:DR: Yes, OP is an insecure douche because him and his girlfriend "mutually decided" to get rid of Facebook.