I imagine the girlfriend wants to be at exhibit nights, etc. with her boyfriend, even if it's an event for his kids, because she is now a part of his life, and the kids are his, so they are a big part of his life. Eventually yes, any boyfriend or girlfriend will be accompanying their new significant other and their kids for certain occasions, to show support to their significant other and the kids. I do think the transition from "friend" to "girlfriend" all taking place in 4 months may be too fast and confusing for the kids, especially if he never clarified that things have moved on from the friendship level and they are now dating. It can cause confusion for the kids, who may be thinking that making out with friends is appropriate and normal. I think he should have said "Guys, remember Christa? I hope you liked her, because I have asked her to become my girlfriend and she accepted."
I do find the bed sharing thing to be extremely odd, I would not do that, even if the man is my boyfriend for a long time, and making out while in bed and the kids are squeezed between us? No, just, no. Kids and adults should have their separate beds after a certain age. They are both entitled to their own space and privacy. I guess if I were to introduce my daughter to a boyfriend, which may eventually happen, I'd wait a while to see if the guy is boyfriend material and may stick around for the long run. Then yes, I would introduce him as a friend, so she has some time to warm up to him. The friend would join us for dinner etc., he would not be sleeping over or walking around the house in his underwear though, nor would there be fondling, foreplay, etc. For him to sleep over around my kid, I'd want to wait quite a while.
There is no reason why the girlfriend could not just sleep over when the kids are in your custody, and whenever it's his turn, then he ought to tell the girlfriend that they cannot sleep together that weekend when he has the kids, but she is welcome to come to dinner and a movie with them. They can sleep together on the weekend and weekdays when the kids are at your place. She isn't going to die if she sleeps alone 1-2 nights a week, or doesn't have sex on those days. I'm sure she managed sleeping alone just fine until your ex came into her life.
I'd be understanding and willing to accept such an arrangement for some time if I were dating a divorced man with kids. I would not pressure him to introduce me as his girlfriend and shove the fact we sleep together down their throats. Kids can become attached to a new girlfriend, or on the other hand, may also become resentful of someone entering their lives so quickly without getting a chance to warm up to her or voice their opinion. Making out with dad like that may make them think she is a floozy. Unfortunately, it's too late to establish that boundary at this point. Going to a therapist with the kids and clarifying any questions or confusion they have at this point would not be a bad idea.