My partner have a slightly larger age gap compared to you and your partner (I'm only a year older than yourself), however I honestly believe that age gap plays a significant role in regards to how children are raised. My partner and I don't often see eye-to-eye in a lot of things when it comes to us raising our children and how to go about raising his children to a previous marriage.
Your partner seems quite old fashioned in his thinking, this no doubt has something to do with the environment he was raised in. My partner comes from a family where the mother stays home, raises the kids, cleans the home and cooks the meals while the father works all day and does not help. While my partner believes that he needs to do his share of the workload, a lot of men his generation do not and your partner might be one of them.
I noticed you didn't mention anything about yourself working, I assume that means you're a stay at home mother living off the one wage which is obviously his? I also interpreted that your daughter is not his, correct me if I'm wrong?
In regards to him not helping with your baby at night, I can totally see why. Your daughter is still very young and 100% dependant on you, not him. She not only needs you, but wants you. Fathers, biological or not, often feel rejected because he is not the person that the baby wants. Your partner might be feeling that because he's not the biological father, that he should only support you from afar and not be very hands on. He also might not know much about how to interact with your daughter... I suggest that you have a conversation on whether or not he wants to help raise your daughter or if he's comfortable with where things are at now. If he does, introduce new things such as him helping out at bath time and expressing milk into a bottle or formula in a bottle and him feeding.
Personally, you seem quite young and naive in the way you think about some things which is completely normal as you are a teen mother and still new at motherhood. Getting involved with an older man and one whom already has children is something I believe you still need to adjust to. For me personally, I've had 2 and a half years to get used to being with one. If your daughter isn't his than obviously your relationship is still quite new which explains why the two of you haven't quite fallen into a family routine as of yet. There's plenty of time for that though!
In regards to you caring for his children, it sounds a bit like you haven't 100% taken on a mother role to his children which is no doubt because they aren't your own kids. This might come off as a little rude but it's in no means meant to, a question I have is: If these were your own kids, would you feel like raising them is being a baby sitter? At this point in time, you caring for his children is the most convenient and cost-friendly option. They could go to daycare but seeing as you are already at home caring for your daughter it makes sense you would care for the kids too.
Unfortunately being a mother does mean that you don't get to go out much because your life revolves around the children. I understand the concern from your partner in wanting to know where and why you're going out, he probably doesn't realise he's coming off as being controlling, maybe mention it to him. You could try organise grand parents / family or the birth mother to take the kids off your hands every so often so that you can get some alone time with you and bubs and not inconvenience your partner.
For your partner's gaming, gaming is his method of relaxing. My partner plays soccer and takes the kids to the bike track as a way for him to unwind. He plays games on his iphone and it frustrates me but you adjust to it. I get the shits with my partner for bringing his phone to the dining table and playing during dinner. Your partner works all day and the few hours he plays games for is like you taking a few hours break. Yes his is daily but you need to factor in the fact he needs that time to relax. I suggest you organise a few 'family' events here and there that he might be interested in... maybe go to a monster truck show, the ekka, a fate of some sort, bike riding etc. Using his kids interests to get him out of the house (which get's YOU out of the house) is the best option and it will help you to bond more with him and the kids and become more family-like.
For his daughter, his daughter probably doesn't like you for many reasons. The fact you're not her mother but she has to treat you like it, the fact you have a young daughter will no doubt also threaten her position as the 'daddy's girl' and also the fact you have her father's attention and not her is something she has to deal with. Try doing small things for her like if she has friends over, bake some cute cupcakes with baby marshmallows and edible flowers etc. Involve her in stuff she likes to do that you can do too, if she likes sweets and treats, make toffee together or if she likes to play sports, throw or kick her the ball in the backyard etc. Take her to a cafe and buy her a hot chocolate and a cookie of some sort. Having 'girl time' can be fun and aid in becoming a family. Pick up a few knick knacks you see out and about that you know the kids will like and insist on your partner taking care of one child while you take the other out and spend one on one time with them.