Not to be a douche but 85 pushups in a minute is just stupid, I couldn't see anyone doing 85 good pushups like that. I'm assuming you normally do solid pushups, so nothing held against you it just seems like a horrible way for your school to measure.
I'm not sure how true this is but I've heard if you can bench your body weight 10 times without much difficulty that you're about average for you're size, I don't do weight training(I really should) but it seems like you're doing pretty good.
By chance how long before a run do you eat / how much water do you drink while running.As far as being athletic goes puking after a 5k is pretty bad but if you exercise often I think the issue might be that you don't drink enough before you run so your too dehydrated during the run leading you to drink too much during it.
If you don't already start more drinking water the night before you run until about an hour before and only sip on the water during the run.
Exactly, I could do 500 pushups a minute if it was only moving an inch back and forth. There is no way 85 full stretch and nose touch to the ground pushups with a flat back can be done in a minute. Even if a guy basically flopped back to the ground from a fully outstretched pushup, it would take more than half a sec with gravity.
Anyway, OP you definitely seem in good shape for your age, so good luck