Yes you can be home educated up to 18, although if you were in school before it might be tricky in terms of convincing LEA that you’re being properly educated not just dropping out - otoh lots of LEAs don’t really bother with home edders anyway so you might be ok.
However - think carefully and maybe don’t rule out going to college - I did GCSEs at home but I think I would have struggled with the A level requirements without the support of teachers at college. I do know of people that have done it though.
Re prices, centres tend to charge extra on top of basic exam fees, for invigilators etc, so my GCSEs for example cost around £100 each