Weepy eye & heavy breathing - allergies?

I'm cat-sitting for a friend who is on vacation, and I have a question about his cat. Aku is almost 1 year old and the poor guy has always had a constant weepy eye (almost like a tear sitting in the corner of his eye; the liquid is clear). I've noticed that he often has sneezing bouts (no mucus discharge though) and sometimes when he plays rough, he has to stop and catch his breath. When he sleeps, his breathing is a bit heavy and his stools are a little soft. But, other than these symptoms, Aku seems very healthy and extremely happy. He has bundles of energy.

I don't believe he has a URI or herpes as he doesn't have mucus discharge, hasn't lost his appetite and doesn't have any symptoms such as mouth ulcers. I'm more apt to think it's allergies - my friend is a bachelor so his apartment isn't as clean as it could be and it's carpeted (which I assume doesn't get vacuumed very often either). I think his loose stools could be due to not having the best quality food.

Any thoughts? (FYI, I read through some other threads regarding Lysine, and I'm going to begin giving it to Aku tonight to see if that offers his eye some relief.) Also, should I be hesitating to bring Aku into my home with these symptoms (I also have an 11-month old cat)?

First thing - and nothing to do with your problem...Aku means I in malay!

Now, basic question first - is the cat a persian? If he is, then the weepy eyes and difficult breathing is to be expected, more or less. If he isn't then, I would suggest a visit to the vet, it may be something else. Either way, I would take him to the vet.

I wouldn't take him home until after the vet visit.