I think it is wonderful that you prioritize spending time with your daughter. She is getting old enough that you can start to include her in chores around the house. My kids loved doing the laundry (usually helping put clothes in) and cooking (pouring ingredients in). This helped them to have special things they did with mom. It takes a bit longer to do with their "help", but it has really increased my popularity with them (we both know our children love us, but it is nice being popular or fun as well). Something else I remember is that when I am gone and dad is taking care of the kids, my kids get excited about me coming home (I just don't see it as much because I'm not there.
In regards to your boyfriend - it is his decision. Just as it was yours to move in and have a child with him. I would suggest that you "make time" to discuss it with him when neither one of you is emotional and you have the time to discuss it a bit without interuption. I have found that if I let my husband know that I want to discuss a conflicting issue with us and then tell him, "I am not expecting an answer right away, I just need to share how I feel and better understand how you feel." (this has been very successful in difficult issues in our marriage) I have found that a lot of times he does not think or progress (emotionally) in an area unless I bring it up to discuss (he is happy with things as they are, until he can see a better place to be, but he doesn't ponder on things like this naturally).
Ultimately it is going to be your choice as to whether to accept where he is. He may change, he may not, but it will be his decision.
I think I would share with him that simply not getting married is not going to prevent the types of dysfunctionality that you two seem to be trying to avoid. Getting married is more than "a piece of paper" although that is the excuse that a lot of people use to avoid it. Marriage (to me at least) is a commitment of a man and woman to start a family and to stick together through all the problems and joys. To put your family as the priority above all else (including yourself, being selfish, self-esteem, and the traditions of your family (including bad traditions like family dysfunctionality). Personally I have had to improve my "tendencies and habits" that I learned growing up in my "dysfunctional" family.
It seems that currently you both are choosing a different path than your parents (in not getting married) but not necessarily a different pattern as far as priorities etc if you both are not willing to put your family above your struggles. I hope things work out well for you all.