Good decisions OP. Sounds like you're making moves. Getting your friend to read through your solicitor's contract seems like a good move as if they've breached the terms, you can (for free I'm pretty sure) raise it with the Law Society. I would argue that as you said in your OP "The solicitor is working on a fixed fee basis so doesn’t want to get involved with any of this" is not appropriate representation and thus must be breaching some level of the contract, unless they have stipulated that they will only do x,y,z in the contract you signed.
Don't pay next terms fees unless the contract is SOLELY in your name. Let them pursue HIM for those. DO continue with finding a new school and please God she'll settle in beautifully. I hope you find a nice school. Long term, are you planning on remaining with your parents? Factor that in when looking at where the school will be - you don't want to have to keep moving schools if you eventually can afford to rent a place for yourselves in a different locality.
I would maybe quit dancing but keep up swimming lessons. I thought swimming lessons are on the state curriculum in the UK? If they are, quit swimming and keep up the dancing. You can also look into free clubs she could join. It's good for children to have extracurricular activities.
Your biggest expense here seems to be school fees.
Once they're eliminated, you need to do up a budget.
Do NOT include ANY money from THE ASSHOLE in the budget. Just what YOU can afford.
Anything HE the prick gives you can be used to pay off debts quicker than you could.
Is he claiming tax credits from you? If so, make sure they're back in your name. Make sure you're getting everything possible as a single parent. Again - please repost on Legal or Lone Parents.
Try not to let the flash cars and houses and shit gall you. You're free now, you've got a beautiful little girl. None of my family ever went to private schools and are all extremely successful. It's not the be all and end all, but change is an absolute bastard to cope with.
I'm sure your dd loves having her grandparents around - all the more people to love her.
You're doing great, but do ask the more specific questions on the boards which might have more expertise in particular areas - e.g. the tax anomalies/different bank accounts on the Legal board; what you're entitled to on the Lone Parents board. Link to this thread too if they need more info, but I think you might get more specialised/relevant information.