Hetero-normativity Reminder: Some of here on GM have GF's and not BF's
I think it depends on the level of the relationship and on what terms you agreed to live together under (was it becoming a family or being roommates). For my relationship my partner does not live with me plan to live together next year and talk about being family. We spend weekends together (it's a long-distance relationship) When we are together, he does bed-time/nap-time when I need to study or stay up for some reason, we equally share things like cooking and cleaning (although I pack lunches for Cae), I do all the toileting, we share bath time, we both read to Cae and we take trade off 'parenting' throughout the day. When I need a break he steps in and when he needs a break I do. We are just now at a point in our relationship (after 1 1/2 years) where I feel like it is okay to ask him (not in a do me a favor way) to do something for Cae, like put on his shoes or put him to bed.
I think for us we plan on being together indefinitely. He knows that if we get married/commit to each other it will mean that he is saying he is ready to be a parent to Cae.
I don't think I could handle living with a bf/gf who was not ready to help parent my kid. To me, parenting is harder with another live-in adult who is not willing to help parent than it is as a single mom with a live-out bf/gf...
SO and I tried living together a year ago and he was not ready and it caused a lot of stress because his presence was a constant reminder of how unequal things were.

My so is not my bd either but we don't live together. We don't have any plans to move in together as of right now but it is in the future, so we haven't talked much about the living together issues yet. So the situation is a bit different than yours. As of now, my bf and my son are sort of at the "friendly" level, where neither I nor my so feel like he should be responsible for discipline (although he does sometimes correct behavior if I miss something). As far as E is concerned, my bf is sort of like his friend. However I do agree w/ Adcaela that I couldn't live with my bf if there wasn't some sort of agreement as far as parenting goes.
As far as your situation, did you guys agree on anything when you moved in? If not, maybe you could talk to him about setting some base rules for how you divide chores, or if you did that already and he's not following or you want to change you could also talk about that too. I think when you're living with someone its important to have conversations like that as often as you need to and also best to do when you aren't already arguing about something having to do with housework or rent, etc.

I'm single right now. I haven't seriously dated anyone since me and my bd broke up a year and a half ago. But I sometimes wonder about what it'll be like.
I can only really offer you the support of knowing that I would feel taken advantage of too if I were in your situation. I feel like he should understand that moving in with you can't be the same as just moving in with a gf who doesn't have kids. The fact is you have kids, they are involved, so it changes things. We don't like having kids to force a change, generally speaking. It's inevitable with relationships when they get serious though. If he wants to live with you then he needs to be your partner in your life. Maybe you two both need to sit down and re-evaluate what it means to be a partner to one another. Make sure your needs are being fulfilled, and be honest with yourself. I sense a probable growing resentment between you two. Those types of things are harder to fix the longer they go on.

I cannot offer direct advice to the questions you asked, but what stood out to me is this: you said "Even though he does x, y, z, I'm feeling like I'm being used right now." So regardless of how you divvy things up, you should explore this more, and see what is at the root of this feeling. It may not be related to chores at all.

My boyfriend an I have been together for a year and a half and he shares equally just as much responsibility for my son as his real dad should. He puts him to bed if I want to go to the gym, etc things like that. He pays all of the rent. but I guess my situations a little different as he has 2 kids that I help with.we take turns doing whatever the other needs at that moment.
imo i dont think thats fair no.

IMO, and this is just my opinion, you do not need to be living with someone who isn't ready to be a family. When you have kids, and you live with someone, you have to become a functioning unit or this kind of resentment is going to happen. Even in my situation with a roommate, when she was treating my son unfairly as compared to her children, when she was expecting me to put in more money/time/etc, it's just not right.
I would not have moved in with my SO if he and I weren't prepared to be a family. When he started shirking on his housework, I spelled it out for him. I told him I was fine living with a person who felt the housework should be more my job (because Matt does cause a lot of mess), but I wasn't about to have a relationship with that person. He straightened up and saw things from my point of view.
My point here is that he needs to decide if he's your boyfriend and partner, or if he's your roommate. If he's your roommate, then it might be justifiable that he gets privacy with his videogames and doesn't have to pay as much rent (though only paying a third seems like he's getting one hell of a deal) but it's not justifiable for you to be cleaning up after him and buying/cooking his food. It's all or nothing. He's a fully functioning member of the family, or he can buy his own groceries, cook them, and stay in his own room and quit bothering your kids.