You can’t warm to everyone. But a lot of the issues with this girl sound like they stem from her mother, and also father to a degree. There is some poor parenting.
At 13 she should be told, gently but firmly, that her questions are too personal. Don’t answer them, and pull her up every time. DO NOT (as a PP idiotically suggested) ask her if her Mum told her to ask X. You already know the answer to that, and this girl doesn’t need to be used more as a political football than she already is.
Hygiene can be hard with some teens and this may be partly her asserting this is her home. I wouldn’t die on this hill because she will get better, but I would try a bit of bribery. If she loves takeaway pizza on a Sat night, make that dependant on a shower and clean hair. Take her to buy a set of shower stuff if she only has odds and ends. Remind her quietly to flush the loo - every time. Ditto with please and thank yous. Keep saying in front of her when DP does something nice. When it’s his B’day etc, take her to get card and present, he should do the same for you. She is clearly learning appalling manners from her mother,
Also it sounds like you are having to do a bit too much - can you reduce it to one family thing with all 3 of you eg movie night, and one 1 on 1 eg baking, and DP do 2 1 on 1s eg swimming and lunch out.
I have step kids, it is not easy. I love mine but not everyone can - the main thing is to do a good job step parenting which you are. Do get your DP to step up a bit.