Hi, I have not read past mid-page 1 and will be responding mostly to the little information given in the original post, so sorry if it doesn't fit.
You have a reactive dog. He does not simply need obedience training, but he needs counter-conditioning. Sounds like the situation can be dealt with a few ways. One being management (tools, avoiding trigger situations and having plans in place for what to do if confronted with on), another being working on the behavior, and the third, well, yeah putting the dog down. Have you heard of Behavior Adjustment Training (aka BAT)? Honestly, it's a difficult feat without the help of a professional to at the very least analyze the situation, but it *is* possible to do it on your own if you are dedicated enough. Fair warning that it is tedious work and is no quick fix. The dog may or may not be able to fully come around in the end, but you can likely get him to a manageable point that isn't exhausting at the same time.
The incident where he bit your mother seems like an unfair thing to base putting a dog down on, in my personal opinion. The aggression would have been redirected aggression. Oh, and the prong collar could very well be exacerbating the issue as it can cause over-stimulation especially in an already aroused dog. Good leash skills can help quite a bit when dealing with a reactive dog. If pulling is a big issue, I'd recommend getting a front-attaching harness. Something like the Freedom Harness or Easy Walk from PetSafe. It will redirect the dog back to you when they pull.
Are you running into these problems on walks or do you live somewhere (apartment, etc) where you can run into these situations just taking the dog out to use the restroom? If on walks, then try to take him during less busy times and be one step ahead. If you see a dog coming up, turn around or cross the street. Muzzle the dog before taking him into any situation a potential trigger could be. BTW, you will want to condition the dog to fully accept the muzzle before taking him out in it. Muzzle doesn't have to be a long-term solution if you go the BAT route instead of management, but for now tons of management is necessary.
Hate to lay it on ya, but you assumed responsibility for this dog. You should at least attempt to work with him on the behavior and do what you can before deciding to euthanize him. If you do re-home, please please please make sure that the new owner fully understands the behavioral problems and will take them seriously. You don't want him to end up with somebody who is equally as unprepared to deal with it and him to get into a situation where he DOES cause serious harm.
Sorry if this is scattered-- low on sleep. Best of luck to you and your dog.