Matt, I woould sugest that if you apply, you ask for a .22 for rabbits, as asking for a .223 for them is going to be turned down flat.
a .243 for foxes will also get turned down on this basis, so apply for either a .223 or 22-250. Both are excelent for foxes, although I would say dont buy a used 22-250.
As for storage, this can indeed be kept at home and the all-in-on cabinets with a lockable inner section will be fine.

It also depends on how big the farm is, you wont get anything on an allotment in Peckham for instance! Although they are not supposed to take this into account, most police forces do.

As for how many rounds to keep, I would apply for the following:

.22 - Keep 600 buy 500
.223 - Keep 60 buy 50

Obviously if your wanting to join a rifle club, you could up these figures to suit, but dont go overboard.

The most important things (as you will read in many posts on here) is not only join the BASC but also ASK your local Firearms Dept for advice, you will find that forming good relations with them is a valuable exercise.

Also worth thinking about applying for your shotgun licence at the same time.