"Welcome to the wonderful world of owning a shit-head corso." LOL!!!
My thoughts:
Q1: My first thought was that you hurt him when you picked him up off the bed.... easy to do... poke a finger in the wrong spot or tweak a limb... not unheard of...
A puppy reacting with a bite in that scenario could just be trying to communicate with you, "STOP"... but they do need to learn that teeth on skin is NEVER ok (at least not in our house it's not). I'd yelp and then give them the command to get off the bed instead of trying to use physical force. So he knows first that the bite was a BAD thing, and two, that he STILL has to do what YOU want him to do... the bite does not END the discussion, nor does it sway your opinion of what he WILL be doing. Bites do not NOT win an argument with a human, ever. In fact... they tend to LOSE them around here.
Some options -
(1) As noted above, remove his access to the bed until he earns it by being a good puppy.
(2) Teach him "UP" (or your preferred command for "get on the bed") and "OFF" (or similar, for "get off the bed")... use rewards for both UP and OFF, repeat this 10-20 times in a row (which is good exercise for the puppy, too!)... until "OFF" is an automatic muscle response to get off the bed... then you can Test him... let him up on the bed, play with his feet, his ears, his belly... act like you MIGHT push him off the bed, and if he resists... command him OFF. Enforce him getting off... tell him "good off" when he's on the floor... invite him back up and repeat the test until he can "pass" without resistance. I call it "play with a purpose"... It's all a game until the human gives a Command... and as long as the puppy does the command as requested, play can resume.
Q2: At the vet... that could easily have been the vet missing the puppy's signals to "please slow down, I'm nervous"... grabbing a muzzle is never something a dog likes a stranger to do, so HOW the vet went about it would really be a determining factor. If the puppy remembered getting the nose juice shot from a previous visit (I imagine that's a memorable event), that wouldn't help the situation, either. But, again... teeth on skin is still a big No-No, especially at that age. So more training and prevention for the next vet visit is warranted.
We were still fighting the desire to use teeth and mouth on humans at 4 months... Denna didn't give up mouthing on people until she was 8-10 months old (her's was mostly in play-mode, though). SO... this isn't necessarily a view into where your puppy is headed, it's just that your puppy has not learned to "use his words" yet, and is still grabbing things to get his point across.
Next vet visit - either have him muzzle ready (mainly so the vet can relax around him) - or take a plush toy that you can stuff in his mouth if you see him opening it to grab anything or anyone with his mouth/teeth. Try and ask for a different vet, or let the vet know you've been working on the issue, and ask him to work with the puppy more before going into the full exam, or at least go slower... maybe someone else has some other recommendations... Denna is also nervous around our vets (we see 3 different ones in the same practice), so I obviously haven't gotten this one figured out, either. Some vets (including one we don't see anymore) are just afraid of big dogs and/or bully breeds... so make a note and avoid those if that's the issue. No need to add their preconceptions to the party, they certainly won't help the issue.
Q3: Puppy social... 15 weeks seems early for him to stop being interested in puppy play... but maybe he's "done" with strange dogs. It can happen. I'd still take him, so he has as much time around other puppies his own age as you can give him. He can at least have more time to learn not to be fearful or aggressive around strange puppies/dogs.
During the social are you allowed to play/interact with the puppies? During some of Denna's puppy play groups - especially if there were only 3 or 4 puppies there - the trainer would get out a tug toy, ball or a fetch toy, to get the puppies all moving and interacting together... it did offer a potential point of conflict, but that was part of the lesson plan - get the puppies to learn how to peacefully resolve conflict with other dogs and/or get them to know when/how to just walk away if they didn't like the energy (versus attacking the "bad" energy). If your socials are led by GOOD trainers, that could work... if they're just puppy-free-for-alls... probably not a good idea to instigate conflicts... so maybe just work your puppy on obedience drills around the other puppies, so he knows he needs to listen in that kind of distracting environment, too.
Keep at it!
If the puppy isn't interested in socializing with other puppies or dogs... keep up socialization by introducing the puppy to new things, noises, location, surfaces (bark, rock, bridges, etc.), and other "stuff".... umbrellas are a good one - closed and open... and the process of opening them, too!