Yes she should feel at home, but she deliberately ignores the rules in which I should also feel at home. I wouldn’t dare enter a home and disrespect what was asked of me, but perhaps that's just me.
OP with kindness she is NINE years old. She has been through a parental breakup, she’s having to navigate two very very different households and she has a tricky stepmum with some quite severe mental health issues.
It’s not your fault you are ill of course, but neither is it her fault she’s having a tricky childhood. What’s more, you chose to become part of her life, she did not chose to be part of yours.
It’s of course fair enough to expect shoes to be taken off, hands to be washed on arrival and before meals, and above all, washed after going to the loo.
However, given her age and the fact her mum’s house is much more lax, it isn’t surprising she forgets. Get your DP to work with her on this. At least she admits it when she hasn’t. Some nice hand soap and shower stuff would be a good idea.
She will be far far more aware that you are watching her than you think - given your condition, you are not going to be able to hide your tension. Sometimes I am sure it’s fun to wind you up - she is just a kid.
I think you need to be realistic with yourself about whether this relationship can work. I am a stepparent so I know how hard it can be, but you do have a duty to do all you can to build an affectionate relationship with your step kids. If you can’t the best thing to do would be to bow out of her life and for your DP to return to his own place.