Unless she was exceptional in other ways I would not be happy with 1, though I'd be OK with her showering etc. after work. Turning up in her dressing gown is not on though.
2 I wouldn't care about.
3 other than occasional, unavoidable incidents is totally unacceptable and I issue written warning.
4 is weird if she has a room for herself unless your bed is particularly large. I would expect this to be just something I asked her not to do if it bothered me. If she did it again without good reason after that I'd be quite concerned.
5 I would be quite grateful if she was happy to take them in her work time, but you say D.C. Should have been at school? Which totally changes it and I think that's a matter for a written warning.
- if that was a one off I'd have talk with her. If it happened more then written warning.
She sounds too immature, unreliable and irresponsible to have charge of children really. But have you talked to her about any of these things? She may think you are OK with what's going on. You talk about her as a live out au pair, and if she were that I'd find a lot of her actions a bit more acceptable really. So maybe she thinks you're OK with it? If she's got another job too she's probably squeezing this in around it and trying to have a life too.
On the wage side, the overnight is too low. (It doesn't seem reasonable to me that the minimum wage for an on call type sleeping over job should be the same as a minimum wage job stacking shelves in a supermarket overnight, but you've got to stick to the law). £10 net is OK, if it's in line with your area but I wouldn't expect someone stellar for that wage anywhere. Childcare is expensive, but it's a job with a lot of responsibility that requires significant skills if done well. BTW how are you paying her net if she has another, main, job?