Well it sounds as though he is getting enough exercise, it should be roughly 5 minutes for every month of age (he is 10 months, so it would be 50 minutes of 'conducted' exercise, i.e. brisk walking/running) a run in the back garden is no problem.
Bite inhibition is quite an important part of his training, it will give him an idea of how hard he is biting and what he is biting. The idea is that you start off training him that biting is OK, then you slowly change tactics a bit and when he bites too hard you shout 'ow' and that ends the game, he will soon start to bite a little softer, you basically fizzle the biting down to the slightest touch of his teeth on your skin/clothes means you shout 'ow' and the game ends, eventually, he will be very careful where he puts his teeth. It does take time, I would guess from a young pup maybe a month or two (with on going training after that - pups cock up every now and then ) but since your pup is a bit older it may take even longer. The main thing is to be consistant, it is no good trying to train him not to bite you, and then kids/DH have a play about with him, teasing him till he snaps, it needs to be everyone involved and everyone knowing what is expected of the pup.
Go and sit in on some sessions of puppy class's before you take the puppy, always go for a class that only uses postitive reinforcment (i.e. treats, praise), never take him to somewhere where they advise hurting or scaring the puppy into doing things, especially with a Staff. Try to pick the class with the biggest range of dogs, big, small, differing colours, this will help with the socialisation. And if a class seems to be utter madness, with dog's jumping everywhere, owners sat having a chat while they should be watching the dogs, don't bother with it, it isn't going to teach your dog very much.
Food wise, the quantity really doesn't matter, it is more about how he feels to you, as I said you should be able to feel his last rib, if you can't then cut his food back a bit, if you can feel more than the first 2 ribs then give him a bit extra. Don't forget to keep checking him though, dogs loose and put on weight very quickly, the last thing you want is a fat puppy, it will do his joints no good at all.
The time he spends alone, is he in his crate?
And the mouthing will ease and finally stop with the bite inhibition training. Your trainer (which ever one you decide) should be able to give you much more info on the actual training.