I think the bottom line is that its hard raising kids. Alot of men dont realise this - Ive no idea how long you've been divorced, but I do believe that alot of the minutae of day to day life with them is alot tougher than many dads realise, because mums often just get on with it, and alot is mental load which is invisible and keeps the ship running.
Of course Im generalising here. But it does seem many times that dads just didnt realise how hard work juggling job, home and kids can be, when they end up with them 50% of the time (especially when they may have done far far less while they were married) and in looking for a girlfriend they are also hoping to find someone to lighten that load - something that is far less expected of a new man in a divorced woman's life. Even on this thread you can see many women who think that it's your GFs place to be much more involved. Society expects women to be caregivers, and women themselves are conditioned to be nice, to help out, to be kind and play a mothering role.
Now Im not saying that this is whats happened here. I do see that you do alot, and would appreciate a bit of a hand now and then. I agree with those who say that the football and trampoline watching and waiting is beyond the remit - it IS boring. TBH I wouldnt do it, and it doesnt take two.
However she IS doing some stuff - Im assuming that if she is cooking on a night when the kids are there, she isnt just cooking for you and her? She is cleaning all the house apart from their rooms when its her turn - so shared spaces that they have been using? As others have said, kids rooms are not easy to just hoover, it can take three times the time just to get all the crap off the floor.
So given what you were having to do when she didnt live there, you are getting 'extra' - help with housework, help with finances, company and shared time when kids in bed / with their mum. She is being nice to them, just not doing the parent based responsibilities. On her loss, as dating someone with kids, she doesnt have you to herself 100%, so she makes sure she occupies this time with other stuff, rather than doing those things during time she could be spending with you. I think thats why she is being straighforward and communicating well asking about times and schedules. She has probably seen too many women finding themselves in the default parent role by stealth, so is making those boundaries clear, as she is only your girlfriend, she is not a step mum, and may not actually want to be one. Its quite an investment of time and commitment just as a girlfriend.
In regards buying - you can get a deed of trust for relative shares in the property, reflecting proportions put in / paid, if you wish to buy together. Of course, it being only your property and her having no stake / chance to earn equity leaves her financially very vulnerable, and if she is clear on her boundaries around childcare, she probably will be as equally financially astute. If you broke up she could find herself homeless, and with nothing to show for having paid into, and helped run, a property only in your name. I suspect you will have some big conversations ahead.