”Her boss has had her working the floor as though she is fully qualified, giving her most of her work. She doesn't get paid extra. Is this allowed?”
Unless her contract says that if she does different duties she’ll be paid more then yes this is allowed.
”She used to pay her travel to college and has now stopped without notice.”
This could be a contractual benefit, either because it’s in the contract or by custom and practice if it happened long enough and became established. There’s certainly no obligation for her to do so though.
”Making all staff take her chosen dates as holidays?”
Perfectly fine. As long as staff get their minimum holiday entitlement.
”Doesn't allow full weeks holidays ie 2 weeks but will give 10 days?”
Again perfectly fine.
”Cut their days without notice?”
Depends what their contract says about hours.
”Is my daughter allowed to do homers outwith the salon clientele?”
That would definitely be a conflict of interest! I would absolutely expect a contract to state that this is not acceptable. Especially as it sounds as though she was actively publicising this, rather than just doing it for mates or whatever.
”Do they stay on the minimum wage for an apprenticeship when they move up a year?”
If they are 19 or over at that point, they have to move on to the appropriate NMW rate for their age group.
”Gave her a new contract this year but wouldn't give her a copy of the last one?”
Well it would have been helpful for her to also provide a copy of the previous one but it’s not unreasonable of her to assume that your daughter has kept her own copy! If necessary, your daughter could do a Subject Access Request under the Data Protection Act 2018 and obtain a copy that way. But that will probably be inflammatory. Is there no chance your daughter might be able to locate her own copy at home in order to compare with the proposed new one?