1. Strictly speaking you would need your original land registry maps and a good engineer to establish whose is the hedge- is it on your land, is it on your neighbours land etc. If it is on your land your neighbour has no right to touch it and vice versa. If it is a shared hedge then they should get your permission to remove it entirely.
2. There is a big crack in the wall- but presumably it is fine for a garage- is it? Is it something that would need to be touched if you weren't converting your garage? I'd say if you want to convert your garage and you need to rebuild this wall you should approach your neighbour, ask their permission to do this- pointing out that you will be perfecting it and only then go ahead.