I vaguely assumed she was doing A levels or some other course, and that he was probably doing construction studies/apprentice day release or something in that vein.
I don't know about family wealth. Obviously it wouldn't be mentioned in the books, as it would destroy the 'alone and destitute' narrative, but nothing of the tiny bits of information we do get about his earlier life when they were first together suggests much wealth to me? Apart from anything else, I assume her parents would have been less unimpressed by him if his family were wealthy, even if he was a 'townie'.
They both seem to have been living at home when they were at college -- in one scene in TWS they are in Moth's bedroom, skipping college, having clearly just had sex, after only a few months together. He waves topless out the window at 'the old lady over the road' of whom he says 'I've been waving to her all my life', which suggests the bedroom is in his family home, rather than a shared/student house.
We know he doesn't drive initially, as it's one of the criticisms her parents have of him, and they drive out to the Roaches in Raynor's 'tiny, battered Fiat.' She also says 'neither of us earned enough money to travel abroad'. W In literally the only mention of Moth's parents I can think of, she says his father drove them to the railway station when they went to camp in Scotland, and was aware of the lie Raynor has told her parents (that she's going on holiday with Moth's entire family, and will have her own room), because he asks what he should do if her parents phone the Walker house. He's described as wearing a flat cap.