I think your role of father comes first, if your daughter is not ready it will only causes probems. If your daughter is not ready to see you with someone else it will lead to resentment. As far as your girlfriend, I think she is pushing something she shouldn't be, especially, expecting to sleep over as well. That would definitely make your daughter uncomfortable. I would choose to wait until my daughter is ready to meet your girlfriend otherwise I think you are looking at a lot of resentment on the part of your daughter.

If your girlfriend has trouble only seeing you every other weekend she needs to get a hobby. She is trying to put her own feelings above those of a child. Does this sound like good stepmother material to you.

I've been on the girlfriend end and we had an accidental meeting, very unplanned and caused us endless problems, especially when her mother decided she didn't want custody anymore and the daughter moved in with us. It killed our marriage. I feel it would have been different if we had met under other circumstances.