There is no rule book for step parents is there?
Everyone knows what is "expected" of a Mother but what is expected of a step mother? You only have to frequent these boards to realise that there are some massive differences in peoples opinions and attitudes.
Personally I think you have to go with your gut. The OP's DSD is living with her a lot of the time so OP is a role model and a big influence in her life, however, she is not her parent or her mother.
I think you need to find a place that you are comfortable with, not what your partner thinks you should be.
My DSD lives with us and only sees her Mum one night a week. I don't try and be her Mum, she has a Mum, but I am her main female adult role model and I try and treat her fairly, but ultimately she is not my responsibility. I don't love her as my own child and doubt I ever will. That doesn't mean I don't want the best for her and want her to thrive. I do and I want to help and support my DP to help her acheive that.
When my youngest DSD has been ill (who lives part time with us), I have helped DP with her, but I let him take the lead. So he might do the clearing up while I sit with her, or I might get a glass of water while he sits with her. When I can see that the situation doesn't need two people, I retreat and let DP get on with it. I would expect and want him to do the same for my DS and me. Personally I don't think a mildly sick child warrants two adults fussing about.
I too share your feelings on being expected to look after a child when the Mother is sitting around doing nothing and I am expected to give up my time for the child. I have felt really resentful when that has happened in the past. Fortunately my DSC Mum is now more keen to pull her weight now she has changed boyfriends so it's not really an issue anymore. I also think as time has gone on, I have got more used to having the kids around and they are older so don't cause me much hassle - teenagers don't demand my attention unlike younger ones.
I think a lot of people see "working from home" as just sitting around watching daytime tv and therefore think they can decide that your time could be better spent. If your partner is one of these people, don't give in. Your work is no less important because it's from home.
As for the playing of sport, it's a difficult one, because he has his DD more than half the time. He is still entitled to have hobbies and a life away from his DD, however, it's how that is acheived that's a real dilemma. A lot of single parents with no family support will find it difficult to participate regularly in anything. Do you think your partner could ask someone else in the family to look after your daughter sometimes or could she go to an after school club and he can play the sport then? If you weren't asked every single time, would that make you feel better?
I think it's probably unreasonable to never expect to look after your DSD alone if she is with you for so much of the time and I also think it's important for her development that she sees her Dad being independent from her, so that she doesn't think her Dad's life revolves solely around her. There are a lot of children parented by "Disney" dads who end up very entitled and arrogant, so that's something to be avoided, I think.