
You're right, it is a lot of changes for him all at once, and it sounds like he is mainly unsettled. First off, DO use food for training, all the time. There is nothing at all wrong with this, it's what all good trainers do. Look at it this way: what would you prefer, to have to work because somebody told you you must, or to work because you would get paid for it? :) Just carrying out a command for no other reason than having been told to gives the dog no incentive at all unless it is a dog that loves to work so much the work itself is reward enough. (But even working dogs like police dogs and similar still get rewards -in the form of play with a favourite toy- after carrying out tasks). You can over time cut down on the treats, or replace them with something he likes as much -even really good praise, but to start with I'd make sure to use food all the time as it will give you an easier life and a happier dog.
It is fairly common for Staffies to be dog aggressive and as this one has had a bad experience as well it's understandable. I'd recommend using a headcollar for walks, that way you can turn his head away from other dogs you meet, as if they don't get a chance to eyeball each other, chances are there will be much less barking, if any at all. The pulling will also get much better. I realise it will be difficult for you to do complete with a pram, but ask your boyfriend to get your dog's attention with a treat when he sees another dog, BEFORE he starts barking, distract him with treats and feed him when another dog walks past, so that he gets rewarded when dogs appear and he doesn't react. Don't reward him for looking at the other dog (and NEVER punish in any way for barking a other dogs), but for looking at the person who is on the other end of his lead. The idea being in time when he spots another dog, he will switch his attention to his owner as he knows a nice treat will appear.
Not peeing at the terrace is probably because it is new and different and he doesn't realise he is meant to. Peeing and pooing when left alone and at night sounds like separation anxiety -was he used to being lelft alone before your boyfriend moved in with you? It all comes down to insecurity. Could you put a baby gate on your bedroom door and put his bed outside so he is closer to you but not in the bedroom with you? (I am assuming the baby is in your bedroom.)