First, I'd recommend that you stop feeding any toppers or feeding any treats if you're truly trying to isolate if something is causing an issue for your dog. It's impossible to know what "thing" is bothering him if you are feeding multiple items. Sometimes even a small topper or treat can be the issue.
As Mel said, some parasites like giardia can be hard to diagnose.
You could switch to another food. Royal Canin and Eukanuba have both been successful for other dogs here (and with friends offline) for dogs that have been really sensitive. If you switch, feed ONLY the food. Use only that for treats. You can't know if it's successful if you feed multiple things.
You may want to also ask your vet about using Tylosin powder for a few weeks to get him stabilized.
I wouldn't personally worry about puking twice. Mine do that on occasion. I don't really worry much about dogs eating fast - that's normal for a dog. I wet the food down with a little water to make it kind of soupy - that seems to help somewhat and it has made a difference for my dogs. I don't use slow feed bowls. I do occasionally use food toys, not to slow them down, but for mental stimulation.
YES! As I was reading the information on what exactly your puppy was getting my first thought was that you were getting too many different things into his "regular" food to tell if he was really reacting to that or to some of the added stuff.
If the things you were adding were to get him to eat well you can do that with other things that don't involved commercially prepared thing which tend to be overly rich, overly high calory and often not the best thing to be feeding a puppy.
I'd try feeding just his kibble--moistened with warm water for starters. (Personally I'm a big fan of Purina's Pro Plan Focus line--but not all dogs do well on the same thing). I don't feed anything that's lamb based--it has never worked well for my Dobes. I prefer not to feed beef based just because I've had several dogs who didn't do well on it. So I choose to feed either fish based (and I prefer salmon) or chicken based and that's at least partly because of digestability ratings.
But I've fed dogs of mine other things at times. Royal Canine has a prescription line that is excellent for specific dietary problems. Never have fed their regular line kibbles so I don't know about that.
Eukanuba/Iams--I've used some of Eukanuba's puppy formulas and their adult formulas for some dogs--they did well on it.
Hills prescription foods are excellent too and I know enough people feeding various of the regular line formulas to recommend them as well.
But ProPlan is my go to as far as food goes.
And I generally feed additives--my puppies get a tablespoon of yogurt or cottage cheese with breakfast and my adults get that too. And my dogs get (adults) a hard boiled egg with dinner (puppies get 1/2 an egg for the first few weeks) and I add meat to dinners too--I mostly don't use canned food as some puppies really don't deal well with it and it makes their stools soft but I can buy chicken hindquarter and cook them until the meat is falling off the bone--removed the bones and chop well or grind the meat and freeze in one pound containers. They get a couple of spoonsful of that with dinner or if they are still eating three meals a day the egg goes with lunch and the meat with dinner. Or if I find a hot deal of either ground beef or ground turkey I'll cook that with a little water into sloppy joe consistency and freeze most of that and give a couple of spoonsful with dinner.
Along with regular food I don't give many manufactured treats unless I was the one who manufactured them. I use string cheese for high value training treats or I have recipes for several really high value treat but for the most part I use the kibble the dog is eating. One piece of kibble hand fed is a big deal to most dogs.
And my dogs all eat a huge variety of vegetables and fruits that they get (usually older puppies and adult dogs--I try not to give real baby puppies some of these things until they are over six months) but all of my dogs eat lettuce, celery, tomato, green beans, peas, apples, pear, bananas etc--in small bites--generally when they are hanging out in the kitchen hoping for some of what ever I'm making--and all of them think berries (strawberries, blueberries, blackberries, raspberries. But these are the kinds of things they get in SMALL quantities.
Good luck--keep us posted on how this is working out for you and your puppy.