Got kids? I see you say you don't. Honestly, I have two sons that have boundless energy. Bouncing of the walls energy and it is completely normal. what does a parent do? Give them outlets. Go outside with them, ride bikes, run races, get them in sports, go swimming, throw balls, play soccer, etc. My goal with younger kids was to tire them out so they fell into bed and it took a lot of parental effort on my part. But that's parenting. It sounds like you'd kind of like him to sit. :>) Doesn't happen for most boys and your boyfriend needs to be out doing things with him when he has him. The whole pictures thing . . . that's really, just being honest here, so rude. Saying that his son doesn't look like him was really barking up the wrong tree. That's like you are trying to CREATE problems and trying to put thoughts in his head to cause distance and doubt with his child. Your relationship is doomed if you keep down this road.
And just going to be frank here, it's hard to love someone else's kid. To allow our partner to be the GOOD and active parent they should be when it absolutely means they will be not putting me first when it's for a child that isn't mine too . . . it's all just really hard. I don't fault you for that and you are not alone. Not a lot of relationships end up happy when kids from a prior relationship are involved. It may be in your best interest to cut your losses sooner rather than later. The kid goes nowhere and there is more to come with the stress of it all.
And I am not trying to say anything about you . . . I know you probably even have some affection for his child and love this man and have a good heart. It's just complicated when there are kids. His child will be his number one priority.
Oh, and before I had kids, my sister asked me to watch hers for a weekend. After about 10 hours, I was like "heckkkkkkkkkkk no". It's hard to be around kids a whole whole lot when you aren't use to it. Now, I can tune a lot of stuff out and as my kids are my world, I will put on tennis shoes and run around outside with them (when I have a million other things I'd rather do). good luck