This will probably get a little long, but I believe it will be worth it.
Several things I noticed right away. The fact that her father introduced you and gave you his permission to date her says a lot in itself. He ends up watching the child while she is out, he needs a guy in his girls life so he can have his life back. No grandfather wants the grandkids all the time. He already raised his family, now he wants to get back to doing what he gave up. You, or anyone else he can get, is the key to that.
You have been dating a whole three months, and you are talking about moving in with her??? Are you kidding me?? There is NO way you know her good enough for that! You are playing 5-6 nites a week, so the vast majority of the time you two spend together is at a poolroom. How much can you two actually even know each other?? Not very well at all.
Moving in together is a big committment. Moving in with a child involved is a HUGE commitment. Before you move in together and become "daddy", you better make darn sure you are going to stay together! The repercussions on the kid can be devastating for life to her, don't take it lightly!
Get to know each other first. There is a TON you two need to talk about before you get that serious. Right now, she is looking for a helpmate, not necessarily a mate. My own son didn't listen, and while still together, is has not been happy times. Given the chance to do over, he would run as fast as he could. What role will you play? Will you be her dad, or just your girlfriends boyfriend? Who pays what bills, who has what bills now, how does she handle money, what pet peeves does she have, does she want more kids and how many, what about discipline for the child,what role does the child's father have and want, ect, ect. A ton to talk about to get to know each other.
Don't think that you will give it a try and if it doesn't work, she is out. It doesn't work that way in real life. Besides your attachment to her child, which, if you don't adopt, you will NEVER have any right to, there is the attachment of the child to you to consider. If you two don't work out, the child can get pretty messed up, and you might and probably won't ever get to see her again.
I had a previous wife that left me. She left me with her two kids. Meaning, she left, and I still had her two kids. I raised them for a time and got them straightened out real good. Lots of love there. Then, one day she shows back up and takes her kids. Nothing I could do but stand there crying watching the kids screaming because they didn't want to go with her. Never seen them since. They are grown now, and I still think of them often. The pain doesn't go away, you just learn to live with it.
Next area- Unless you are doing it for a living, you have no business being out 5-6 nites a week and having a family. That's not a family, that's a maid service for you. You can't have a family and all the perks of being single, and vice versa. She has to respect your time playing, but you have to keep it in perspective too. My current wife, early in our marriage, started getting upset about me playing so much. And it was helping pay our bills at the time. She knew I was a player, and I even spent the third nite of our marriage (on our honeymoon) playing pool ALL nite with her at my side, trying to make more money.
We started argueing more and more about it, and I snapped and pulled my cue out of the case and smashed it into toothpicks on the kitchen sink. Then, I turned to her, and asked her "know who do you think I love more?" That was all she needed. To KNOW that I loved her more than pool. She spent her entire next paycheck, at her insistence, to get me another cue!
You do have to expect to cut down on the pool. Not give it up, but cut it down. 5-6 nites a week, that isn't a relationship with your girlfriend, that is wanting nothing more than an easy booty call to come home to. Guess what? Pretty soon, it won't be very easy at all! And, you have to expect to put in some time with her child. If you two are together, it is also your child.
I know it's long, but I hope it helps you do some thinking about a very serious subject.
Damn You're a real buzz kill!

But wise