I voted a whole hearted yes but not becasue of the behavior issues. My response came from knowing that (in the US) millions of unwanted dogs and cats are euthanized every year (yes, that millions each year) because of unethical breeding practices or people simply not spaying or neutering. Also, because not neutering can cause medical and behavioral problems. And knowing, as a vet tech for more than 5 years, that the majority of dogs that I say kept intact for breeding weren't dogs that should have been breed in the first place. Horrible temperments, bad breed examples, possible genetic problems they could pass on.
To me it sounds like your dog could feel less "manly" after neutering but I wouldn't count on it, do it, but don't count on it. He may not feel as territorial and yet a truly dominant dog is going to stay dominant. I neutered my dog at 6 months and he is one of the most dominant dogs I've ever met. (He's also very well trained to nobody notices he's so bossy.)
I shall echo the others who encouraged you to start training immediately. If you can't at all afford to get him into a class, get a book (I personally love "Good Owners, Great Dogs" by Brian Kilcommons if you can get your hands on it). Your dog does't currently know how to behave and if you are trying different things he's probably also confused which is going to add to his hyperness. Remember every reaction you have to something that he does is going to encourage or discourage that behavior. If you tell him to sit and he doesn't sit and you throw up your hands and say "he's so stupid" you've just encouraged him to do nothing when you say sit. If he doesn't sit and you stop what you're doing, make him sit and then praise him you've just encouraged him to sit when you tell him to. If he jumps up and you put your hands on him and push him down you've just encouraged him to keep jumping because he got the attention he was looking for. If he jumps and you put your hands in your arm pits and turn around and ignore him, he'll stand there looking confused and then you can praise him for not jumping (this usually only takes a couple of times before a dog figures out my "four on the floor" rule).
Training is going to show your dog the behavior that helps him fit into his place in his pack (your family), everybody will be happier. Maybe once you have the basics down you can find an activity that will help him get his energy out. Agility is a good one. A tired dog is a good dog.
Good luck!