I'll list the obvious stuff as it comes into my head:
1. You can make a living from anything if you're a good businessman. Car sales is no different.
2. Insurance, insurance, insurance. Public liability as well as the obvious insurance for the cars.
3. Yes, register it with Companies House. Depending on turnover you may need to be VAT registered as well.
4. It's illegal to sell cars from home, your local council will have a field day with that. You will need premises of some kind.
5. Anywhere and everywhere. Every car you see is a possible purchase, but you've listed everything obvious. Part-exchanges will probably be a good part of your business I suspect.
6. Read the SOGA and SOGASA, your legal obligations are in there. In short though, a minimum of 3 months warranty will need to be supplied from either you directly or via a warranty company.
7. How long is a piece of string? Too many variables, but on each car I'd be looking for a minimum of 20-30% profit bearing in mind all the overheads you'll need to pay for.
No offence meant, but judging from your questions you're in no way ready for this at all. It's one thing to see the odd car for a bit of extra cash, but another thing to start a brand-new business from scratch which I'm assuming you have no experience of. I wish you luck, but you really need to do a serious amount more research yet.