Look she sounds like a pretty great kid/ her school describes her as a role model and her grades are excellent. That's fantastic. Credit where it's due - and a testimony is o you too, OP.
The social side. Some people are very happy with their own company and only one or two friends. From what you say, she doesn't sound unhappy about the social situation, it's more you who is struggling. So don't make an issue when there is t one for her.
The issue for you however, yea it is hard, undeniably so. You feel guilty leaving her alone all day. I would too. I'd want to know she was having someone check on her or at least to say hi. That doesn't mean it's wrong but I understand that anxiety.
But with regards to your needs, I think it's time to make a poor of doing things for yourself, to take walks alone, to meet friends for coffee, to take the occasional night out. Start now and in a year you'll both be in quite a different place, she will be accustomed to you having your own life and you will have built up your social life again.
Wrt the spending, that does sound excessive. In my view she is receiving a generous allowance. The £500 plus gifts at Christmas is a lot, it won't mean the same to her as you because young people have yet to learn the value of money. Honestly I would reduce it to £150 cash and £150 gifts until she has better money sense.
With regard to her spending all her money on one shot early in the month, this is the hard bit. You need to stick to your guns about not giving her more. Maybe a weekly allowance would be an easier option given her difficulty managing money?
And absolutely no to her sneaking stuff into the shopping trolley. Kids I'll push, they are testing boundaries and she may kick off but that's too bad. No her manipulative ways should not be tolerated.
It's hard on your own and I do sympathise but honestly it will pass. And the work you're putting in is definitely worth it.