Ted was able to hold his own in a fight with John, so why didn't he fight off his kidnappers?
Two against one? Also realize this: Ted is the sort of Jerk Ass With A Heart Of Gold who is comfortable beating the crap out of his friend. But John? The entire story is about John holding on to childhood and that means Ted. John can't win in a fight with Ted, because John would never use any thing close to full strength on his friend. So any fight with Ted and John is a Curb-Stomp Battle because John doesn't have the heart to trying to win. It's literally their friendship holding John back. And to boot, look at their fight in the hotel room. All it really is is the sort of childish roughhousing that kids do. John is basically just grabbing at Ted in anger without really doing much actually harmful offense. Teddy is mostly doing the same - throwing things, grabbing and slapping. It's only when actual harm is done - the TV - do they stop fighting. So at this point in the movie when he's being chased, Ted realizes this - he realizes that he's holding John back - and so he probably realizes that the two villains he's up against don't have the same reserve (they did rip an ear off after all) and they're not just roughhousing.
How come no form of government or scientist considered dissecting or studying Ted? Or why not the public be surprised magic wishes are real?
One of the first things we see after Ted comes to life is a media frenzy. Ted and John were the darlings of the moment. If Ted suddenly disappeared, his absence would be very, very conspicuous. Even months or years after the fact, the Men In Black wouldn't have been able to do anything without calling attention to themselves. And after that much time passed, as the Lemony Narrator pointed out, no one gave a shit.
It's unscientific to destroy your only specimen, so dissection and vivisection is out given that there's only one Ted. They probably did do less invasive tests, and concluded that yep, it's a magic teddy bear with no way of replicating the phenomenon.
Why did Donny get off scot-free? Yes, they decided that kidnapping a teddy bear was an incredibly stupid reason to convict someone, but he commited several other crimes. Breaking and entering (Ted was still inside Lori's apartment when they bagged him. A technicality, but still.), grand theft (Ted is literally one of a kind and a celebrity to boot, he has to be worth MILLIONS as a collectors item), reckless endangerment(See his crazed driving), child endangerment (his kid was in the back seat), and hit and run. There were so many crimes he committed that he could have, and should have, been charged with.
In order for Ted to have value as a collector item, he would had to have been appraised, which he wouldn't ha e had done to his friend. So Ted wouldn't be worth any more than any other of those teddy bears. As for everything else, it's possible he just lawyered his way out of all that since it would be near impossible to prove. Though it should be noted that we don't see his kid in the sequel so maybe he did lose custody.
John is giving his terrible song onstage and the crowd is booing him for it. Improbable, but hey, Rule of Funny and all that. But then someone runs onto the stage to assault John, and he defends himself by smacking the dude with the mic stand. Then the crowd boos John while he's escorted offstage. Where the hell was security before, when, you know, some random dude was about to assault the singer they were supposed to protect?