From gov.uk you can hunt the following legally:
You don’t need a licence to hunt:
game birds, eg pheasants and grouse
quarry birds and certain wild birds, eg moorhens and woodcock
certain waterfowl, eg some ducks and geese
You don’t need a licence to hunt deer in open season. Open season varies by region and species.
There are no hunting seasons for other animals.
You can hunt some small animals, such as hares, rabbits and foxes, all year round. But you can’t shoot hares or rabbits (game animals) on Sundays or Christmas Day in most of the UK.
However all of the above are restricted to seasons, and / or various permissions to cull / control. Details of which can be found on the gov.uk website.
One not listed directly by gov.uk but that are hunted and eaten due to being a pest are Grey Squirrels
As you mentioned rabbits, yes for the most part they are safe to hunt and eat (myxomatosis is still in the UK but rabbit is hunted and sold so just be careful with what you shoot) - my local butcher often has some for sale which are caught locally. But often you need landowners permission to hunt them. Most people shoot rabbit with only an air rifle and the more powerful licence free ones will kill them (and leaves less buck to pick out).
Something to bear in mind is the wildlife licence which can be required if you're considered to be controlling a pest.
There are also restrictions on equipment you can use to hunt in the UK. For example, no dogs for hares, deer, foxes etc - but they can be used to retrieve these. No bows, nets, vehicles, explosives can be used to hunt in the UK.
For some serious detail you can look into the Wildlife Act, which you have to conform to for hunting animals in the UK.