Firstly, all credit to you for not handing him over to a pound where he could well be PTS or to any old interested party.
In my experience, as a dog rescue volunteer, fosterer and owner, and one who has trained a 6 month old terrier out of such behaviour it seems that your dog is taking the pee and testing the boundaries. I would suggest that you lay those down as you doubtless do for your children - i.e. that your kids know that no amount of bad behaviour will get them what they want when you have said no and that you as Mum are in control.
I'd start by being very firm, but not unkind, about pooch sitting on the sofa. If he is growling when you tell him to get off or when he doesn't want to do as he is told and you give in he quickly learns that this is the way to get results and the problem will escalate until someone does get bitten.
He needs to know his place. So, if he is on the sofa make him get off - use a VERY firm tone (sometimes a mans works better unless you have a mouth like mine lol!) and don't back down. A crate is a good idea but for gods sake don't put him in it as a punishment or it will be a place he definately won't want to go into. Introduce him to it with a toy and treats or chewy bone, and leave the door open so he has somewhere to go when family noise and chaos is too much for him - that way he will make it his own.
Each time he returns to the sofa, do the same. Feed him after the family and let him see he has to wait and ensure no-one gives in to his pleas or demands for titbits until you are ready to do so (I confess I'm a beggar for sharing my pizza etc wih my two dogs but only cos I know they will not get assy in other ways and get snappy if they don't get their own way). If you give him what he demands when he demands it he will soon learn, as he appears to have, that being bad-mannered works.
And yes, DO ask your rescue for advice. If they don't help, shame on them! But all is not lost. My own local rescue has a forum - come join us and we will do our best to advise further, the owners included, we're a friendly lot. We can be found by Googling poplar Farm Kennels Forum.
Also you may like to join the forums of other organisations like Dogpages or Biggsd - the latter is primarily for GSD owners like myself but caters for all typed on its forum "doggy chat" thread and again the members will be happy to help you.
Whatever you do please try hard, as you are doing now, before you give up. I see so many dogs in pounds and struggle to save them and get them into no-kill rescues before they are PTS (ref my thread on here "Considering owning a dog..."). I am sure that you will understand where I am coming from if you have the time to read that thread. If I can help further please pm me - or if the pm system doesn't work for any reason say so on here and I will give you my contact details somehow.
All the best!