Most Helpful Girl
Does he give you an accounting of what happens when he goes out with friends? I assume he doesn't. He might be mildly controlling since he doesn't want you going out with friends (asks you to tell them you're busy), tries to screen your incoming calls/texts, and asks for an accounting of your night out. However, unless you also tries to isolate you from family/work then in my opinion he is mildly controlling and it is very likely due to jealousy (you're spending time with people you don't HAVE to spend time with and you might be getting hit on). If this is the case, it is possible that if he works on being less jealous, feels more secure, he won't be controlling. If he tries to isolate you from work as well or family as well, then he's probably actually just controlling and it would be extremely difficult to fix that.
He sounds self centered, he considers only himself. He wants you to stay over (without considering that he lives with his mom, and what she might think of you staying over also what if you had to pick up your daughter from a sitter/family/friend). He likes hanging out with friends but doesn't like you to hang out with your's because he would rather you spend time with him. He doesn't wish you a Happy V-day because it likely didn't occur to him as something to do because probably like most guys, he doesn't care much about V-day and does the requisite date/flowers because it's socially required of him to do so. You feel like an afterthought because he thinks of himself first.
I think it's fair to not involve him in your daughter's extra-curriculars and to not introduce him to your family. I'm really on the fence with this guy, not a huge fan of him based off of what you have said. Assuming best case scenario, he's not controlling he was just jealous and he is self-centered, those aren't easy traits to alter. Do you want to put in the effort to help him fix that, personally I wouldn't because you also have a daughter to raise and work and friends, you have a busy life. In addition, he likes his alcohol a bit too much (is he borderline alcoholic or doesn't have much control over himself when he is around alcohol?), which depending on the severity, I would not consider anything long term with him because I would not want my daughter to grow up with a step father who has a bad relationship with alcohol. If he is actually just controlling, I would leave him, a controlling behavior it can lead to more emotionally abusive behaviors.1THIS IS NOT RELEVANT ANYMORE