Sounds like the older cat is being a little rough on the new guy since he's causing him to "squeel"...at the same time, it sounds like play too since they chase after each other.
Might want to step in a little quicker and get onto the bigger, stronger cat by telling him NO firmly when he does that. Maybe a tap on his butt as well so he'll associate it with "I did something wrong".
I've got 2 male kittens, brothers 5 months old and they can get really rough. I let them rumble until I see them start biting each other...that's when I break it up. If they get real bad and continue with it, I make them seperate. One cat at each end of the couch, but they have to stay apart for a while. Its not easy, but when they hear my voice get serious, they settle right down (for a while
Its the kitten in them and I know they are just felling out their instincts as well, but I cannot stand for one to injure the other. That doesn't work around this camp!
Sorry I can't help you with medical issues...sounds like a cold or allergies to me. One of mine has sneezing fits anytime I change the litterpan...all that clay dust makes he and I both sneeze like crazy!