Hi D.,
Yes it does sound like your child has allergies. I have two boys 3 & 4, they both have allergies & asthma. I give them over the counter Claritin chewable tablets. It helps some. You can take him to the doctor and they can give you a prescription for allergies. My children were give Zyrtec, but I just assume to buy the over the counter medication. That is only because I don't have health insurance right now though. But it works just as well.
Also, I have allergies and this morning my eye was swollen also. So I'm pretty sure that this is allergy sysmptoms.
Good luck and hope it gets better.