Aggression in a 14 week old pup would be really rare. All pups get excited and bark, growl and bite. Puppy bites are hard and they hurt. He has not learned good bite inhibition yet.
Let me try and dissect your post.
"Melo is 14 weeks now and enrolled in puppy class. We have two-ish problems with him right now: excitability and the resulting barking and biting. I firmly believed that proper training would mean never needing a harness or special collar but now I am starting to think i need one of the two... It is very important to me to get Melo on the right path now before he weighs more than I do. He gets very excited when other dogs are nearby and pulls and lunges and barks. If we are on a trail or in a park just walking he is fine, but in puppy class and at the vet he wont shut up and stop pulling. We (me and the trainer) try to distract him with treats and toys but they are only temporary. He is too young for the dog park so he hasn't had any doggy play since he came home at 12 weeks. The trainer is now suggesting I get a front lead harness in order to prevent damage to his neck and help me control him. Would buying a harness be the official sign I am failing at training him? 
Right now when he pulls I stop and wait but he doesn't seem to be dissuaded."
Have you done any training at home? Puppy classes can be great but I feel like foundation training in non-distracting environments should occur first and you work your way up to distractions (puppy class). If you cannot keep the attention of your dog with treats or toys when the trigger for the excitement is around (other dogs), than you are too close to the trigger or pushing him too fast. Your trainer needs to back him up or slow down the training. You cannot take a dog with no foundation work and put them right in front of their trigger and hope they behave. Being excited by other dogs is completely normal however barking and lunging is not and that needs to be controlled right away as he in reinforcing bad behaviors every time it occurs. If your trainer is not able to identify this and work him up to his trigger properly I would think about dropping out of puppy class and getting a trainer to work with you one on one in private lessons. If you stay and every time he is around dogs he is barking and lunging you are making this problem a lot worse. Right now it is just excitement but if it is allowed to continue it could lead to frustration which opens possibilities for aggression. He needs foundation work and you need to get him to pay attention to you when out vs. the other dogs. A good trainer can show you how to do this properly and build up to the trigger.
The biting. There are lots and lots and lots of posts on stopping puppy biting but I haven't been able to get a clear definition of "play" vs "aggression" and how to handle it differently.
It probably is not aggression. Even if it was the fix for biting for aggression vs. excitement is the same. You have to change the behavior to change the emotions. A dog that is overly excited needs skills/foundation to occupy his mind. So basically you have to work on foundation skills to focus and calm the dog. See a trend here? A puppy getting overly excited is normal and mouthing is normal. It does hurt, it can break skin and it can rip clothes. All normal. The trainer you use to help you with the dog excitement will be able to help you with this issues as well. You need to control the chaos. When he gets too worked up he needs a refocus. Doing obedience commands, some games of find it, grabbing a toy and playing fetch or even a time out to change what is going on in his mind. Teaching a place/mat command would be excellent for this issue. He gets too worked up and you send him to his mat to calm down. You learn from the experience and try to watch for signals and refocus him earlier next time so he does not get into that state.
"We played fetch for a little bit but then he started to get sassy..chewing on the long leash he was on, snapping at my hand when i removed it from his mouth, attacking the grass ?>.< and ripping it out in chunks and throwing it around..."
This could be a couple of different things....
He could have gotten too excited to the point that he loses his mind and makes bad choices. He could have gotten bored with fetch and looking for something else to do, he wanted more attention, etc.
Fixes -
-Run him through some obedience commands to get control of him and get him to calm down. Give him something to do to focus his mind. He can't be bad and good at the same time. Switch which mode he is in.
-Throw down the towel, AKA the mat, and send him to his mat to cool down for a minute.
-Turn around and ignore him until he tires of the lack of attention and once he stops immediately praise and give him something to do.
I told him no and removed the big clod from his mouth and he started trying to bite my hand..HARD bites, like how dare you tell me what to do kind of bites. Similarly, on a dif day I grabbed his collar to bring him away from an area we are training him to stay away form and he kept trying to whirl around and bite me hard."
This could have been over excitement or frustration. You have to remember he is a puppy. Think of it in relation to a toddler. They absolutely lose their minds sometimes for no reason. This is when refocusing him is key. Touching a overly excited dog can cause redirection which is probably what happened here. He took all that focus he was using on the leash and turned it on you. Instead of touching him when he is excited it would be better to refocus him on something else. I NEVER grab my dog by the collar unless I HAVE to. If your training him he should be on leash. You should never have to grab his collar when he is in an excited state. I actually have this little mini leash hanging from Bear's collar that he wears all the time. It allows me to grab him if needed without grabbing his collar. A lot of dogs don't like it and it is almost a natural reaction for them to mouth when you do it. When he is super excited grab his leash and put him in a time out. Use the leash to direct him.
Lastly. is he getting enough physical exercise and mental stimulation? This is SOOOOO important for a young dog. Mental games and training will tire out that brain and make him less excitable.
