I think the 10 year old behaviour sounds normal and pretty expected. She's got a lot going on. Do you have children of your own?
I think maybe it would be helpful to slow down the relationship - I know that's not what you want to hear but it would probably be easier to win her trust that way. I appreciate it might not be easy or possible to move back out (and of course that feels like a horrible idea)
DH moved in with me and DS when I was a single parent, after 10 months and it was absolutely too soon. What saved us was that he happened to get a job really far away so we decided to go long distance while he set up living at the new place and then me and DS moved back in with him over there (here!). It worked well but of course that's not always a situation you can engineer or practical or desirable.
A 10 year old is on the cusp of puberty, her hormones are all over the place, if the boundaries/expected behaviour has suddenly changed (even though you as the adults can see this is a positive thing) she's going to find it hard. The younger ones will be better with this because small children feel safer when boundaries are in place, however a 10 year old not-quite-adolescent is concerned with breaking out of boundaries and making her own way in the world and so sudden changes in the goalposts are confusing and difficult for her even if they are making things more stable. I really think teenagers (and I know she's not a teenager yet) do need different parenting from younger children and IMO it needs to be much more flexible. I think that you could start this now and it may have a positive effect.
Of course seeing her dad is going to have a big unsettling effect too especially if he is showering her with guilt gifts and creating an unequal balance. Unfortunately there's nothing you can do about that.
As an action plan of sorts, I would stick with your current approach with the younger ones, but also encourage your GF to gain confidence with this herself. If she's not comfortable doing something she needs to find her own way but if it's about confidence maybe you can help and encourage her. With the 10 year old, I think you probably need to take a step back and let mum do the bulk of the parenting. She's older and not so ready to accept you as a father figure, and that's fine. That's just a part of step parenting older children and you shouldn't take it personally. The kind of "different" parenting I was talking about before is making it less based on rules, rewards and punishments, and more based on rights, responsibilities and relationship. (N.b. - the "you" used after this is general, I really mean your GF as I feel she should be the one doing this.) That is flexible - you can give her more responsibility when she's handling it well and pull it back when she's not. Slowly you introduce the idea of different responsibilities. Maybe start with a responsibility she already has, for example, homework. Tell her that now she's getting older, you feel she's able to take responsibility for herself. So from now on you are not going to insist that she does homework at a set time, or ask about it, or check that it's done. It will be her responsibility to plan her own time to make sure she has enough time to fit in the homework around playing or seeing Dad or social appointments etc. Tell her you'll be on hand any time she wants help with homework or help with scheduling, or if she wants you to remind her or check etc that's okay, but the whole thing comes with conditions - if she gets behind on homework, or is doing it after bedtime or something, then she'll have to go back to having it managed and you telling her when to do it.
Once you've covered all the stuff she does already, you can start adding on things that you currently manage quite a lot for her, e.g. if you tell her when to have a bath/shower, you can tell her she's allowed to decide for herself now and you won't nag her about when she does it. Divide up the "treat food" or junk food bought in the weekly shop and put her amount in a separate box or label it so she can choose when she wants to eat it in the week and doesn't have to ask (but when it's gone it's gone) - may be more of a boy problem! :)
Then you can add some in which are more of an incentive - having a mobile phone if she doesn't have one already, having a better one. Being allowed to go to the shop on her own. Deciding when to go to bed (?) - whatever you feel she's ready to cope with and would be enticing for her. Remind her that you won't micromanage but you can pull the freedom back if she shows you she isn't handling it well.
The last one to add in would be things which help her to become independent. Things like not directly buying her clothes/toiletries but giving her extra pocket money to spend on things like this so that she can make the decision whether to spend more or less and budget. Letting her have an input into choosing and cooking food for the family some nights. Encourage her to take the bus somewhere rather than expecting lifts all the time.
Most of this stuff is going to be inappropriate at 10 and to put all of it on her would be far too much to cope with and manage all at once, but the idea is you introduce responsibilities slowly, over the teenage years, and see how she copes with them. But handing over responsibility for a lot of things means that you aren't engaging in battles over something she would probably make an okay choice about if left to it. It means that the things she's not coping with you can focus on, and it helps her feel valued, grown up and independent.
Obviously the relationship part is really important too. Especially when she has a lot going on, her relationship with mum especially is really really key. That doesn't mean buying a relationship obviously and boundaries are still important, but making time to let her know you're there and listening and you care is important even if she doesn't always show that she appreciates it! Having a mum who listens and doesn't immediately judge or panic is priceless. That might mean developing a tongue of steel for when you have to bite it
but generally sympathy first, communication second, discipline/advice last. Give her space to work things out.
Good luck! :)