I think you absolutely need to take him to a vet (or whoever owns him), and have a checkup especially if this is sudden new behaviour. It could be an indicator of pain (especially the petting or being disturbed while resting), or some other condition causing sudden behaviour changes.
I also do think you should train him not to be on the couch, or to politely remove himself from the couch when you ask. Terriers are so incredibly trainable, but generally do not give a rats ass about corrections. Not to mention, his biting/snapping at you is a reaction to something happening to him, not a behaviour like barking at a window. If he is immediately turning to biting, it means that previously his other signs of discomfort, such as physical body language, or growling, have been ignored or punished, and he has learnt that the only way to get things to stop is to go the full distance to bite.
I would train him to know the words to hop off the couch, using something he finds really rewarding (cheese is a favourite in my terrier household). And also get him a super comfy dog bed that’s his own space, and train him to hang out there instead. When he’s on it, he’s safe from being touched or bothered.
With not liking being grabbed when he thinks something bad will happen, this is honestly a pretty simple fix: just add in a leash or slip leash! I carry one with me day to day as I have 5 dogs and need to be able to quickly get a handle on at least one if they’re up to shenanigans (3 are terriers so shenanigans are a given). If he is fine with have a leash clipped on, just put him on a leash and then take him to wherever you need him to. If he’s not fine with it then a slip leash will allow you to do this without getting bitten, until he learns to trust you more.
As for biting children, I agree with everyone else: he and children should not be around each other. Personally I would pick a room that guests do not readily frequent (like an office, spare bedroom, or laundry) and close him in there, either with a baby gate or just by shutting the door if he’s fine with it. Make it comfy and a calm routine with a bed and toys/food puzzles if he wants them. Make sure the kids know not to disturb him. Lots of dogs don’t love kids, but biting them is where he borders on becoming a dangerous dog.
Biting when you try and take things off him is a pretty classic example of resource guarding. It’s a hard thing to train out of a dog, especially guarding from humans. When my dogs have something I think they shouldn’t have, I always trade them for it with some of their food. Even the dogs that are fine with me taking stuff.
You make sure they’re aware of you having the reward, use a cue word like ‘trade’ or ‘drop it’, and either swap them for the item, or scatter the treats away from the item. This gives you time to grab whatever they had while they’re distracted. After a while they learn that the word means they’ll get a reward for giving up the item, so long as your reward is better than what they’ve already got.
Teaching him a better way to react to things/training him to not be on the couches, is NOT avoiding the problem. It’s addressing it! In a way that’s lower stress to you, him, and everyone else in the household. Being constantly yelled at or yanked around only adds to his stress and his knowledge that he can’t trust the people around him to be gentle with him.
Terriers do have big feelings about things, but they are also so fun to train when you are working with them, not against them. I have never had a terrier in my 15 years of owning them that didn’t respond immediately and exuberantly to being trained in a kind, fair manner.
And when you can, I would definitely seek out a professional that can help you in your home to train both the people and the dog. It will help make your home so much more calm and harmonious.
