a. pay attention when your vet tells you what they are doing... would be useful to know what they put up its nose, thats part of being a responsible owner.
b. ask your vet, not a forum of 14 year olds
c. no it isn't because he has hair in his lungs
which vaccines was he given?
it is best to spread out vaccinations over time instead of slamming a puppy with everything all at once. Some vets still just give them everything at once, and it is not uncommon to see reactions.
It does not sound like your puppy is having an allergic reaction, those tend to show up much faster and they would be causing much worse symptoms.
Find out which vaccines were given, which brand of those vaccines, and check the listed side effects.
Was your puppy given a bordetella shot? (kennel cough vaccine)
or recently boarded in a kennel?
from the limited information you posted, it could be any number of things.
also by three months of age he should have already had all his vaccinations; except perhaps his rabies.
good luck