Your dog has fear issues and maybe some resource guarding. I would consult a certified animal behaviorist (be careful, many people call themselves 'behaviorists' without proper qualifications because it is an unregulated profession) or a positive trainer who specializes in reactive dogs.
It sounds like your poor guy was punished for a variety of behaviors (i.e. fear of confinement, fear of the leash and collar) and is becoming reactive when he feels threatened or nervous. Many little dogs hate being picked up, it dis-empowers them and makes them feel defensive. I would have your mum stop picking him up at all. Stop petting him, or even really looking at him for now. Wait for him to solicit contact with you.
Biting is a last resort for dogs who are asking people (or other dogs) to back off. Watch your dog. Does he lick his lips, freeze in place, look stiff, yawn, show the whites of his eyes, look or curve his body away from you, crouch down as if startled? These are all signs that he is stressed out and telling you politely to back off. It may be that, especially if he was punished in his previous home, he has stopped offering these warnings because he learned that they didn't work. You can start mending that bridge by respecting his space.
The resource guarding (if that is what it is) is a whole other issue and should really be thoroughly explained by a qualified trainer who can see the behavior. Under no circumstances should he be corrected for it, that will only increase it's severity.
Housetraining is housetraining. If he still goes in the house than he either has a medical issue or has not fully learned to go outside (is it urine or feces or both?). The only solution is to monitor him completely (so, he's never out of your sight) and putting him on a strict schedule of trips to the yard. Every 2 hours and after meals.
Is he still intact?
Reward successful toilets with praise and food. Interrupt accidents with a "whoops" and take him outside as fast as possible. I would put him on a harness and have him drag a light leash at all times so that you don't have to touch him directly to move him outside.
Make sure that you are cleaning all the areas he has messed with an enzymatic cleaner - available at mos pet stores. This will de-nature the scent proteins left behind by normal cleaners that your dog can still smell, and will therefore think makes that spot an acceptable place to go.
Finally, I would suggest that you folks stop feeding him from a bowl. Portion out his daily ration and feed him from your hands. If he is very skittish of specific people they can start by dropping kibbles for him when they walk by and go from there. Go at his pace. If he looks or acts nervous back it off a bit.
A little basic positive training (sit, lie down, shake) will go a long way to give him more confidence and strengthen his bond to your family.
hope this helps