For anyone experiencing similar symptoms, gonna post a recap from today's visit. Vet said that it's very likely to be an allergy, since his symptoms didn't really worsen, other than 2 or 3 breathing difficulty attacks (probably due to extra exposure to the allergen). Chances to be a foreign body stuck in his nasal passage or ear are very slim, because he would've already gotten at least some yellowish/greenish, pus like, nasal discharge and a lot more pawing and overall discomfort. Same goes for kennel cough, but since it has a longer incubation period, he said to keep under supervision for 2-4 more days (should at least get some specific coughing, fever or something related to infections).
He said that a blood test would pretty much say nothing in this case, and a lung X-ray is also not needed since he didn't hear anything weird with the stethoscope.
He just recommended to stay away as much as possible from grass for now and keep an eye on him. Prescribed Aeurius in case of other breathing attacks.
All in all, my boy seems to be getting better. Not much sneazing, other than a few times when taken outside. Nose doesn't seem to be as runy either. And he seems to be getting his energy back, as he wanted to play today. As much as I dislike his zoomies sometimes, I 100% prefer that than not feeling well and laying down, heh.