my son is 2 is allergic to mostly everything except for food. Just like the previous post, keep the house spot CLEAN. Wash everything with hot water with sensitive no scent detergent. Don't hang your cloth outside just use drier. Stay away from pets, and try to stay indoor when high pollen outdoor. If child had allergic attack just give OTC allergy medicine till he gets better. That's what my allergic ped told us to do. Don't worry, it does get better. When the kids are old enough I would sent them to get acupuncture, it works wonder for my husband.
Ugh, my son is seven and also has allergies such as you describe. Tips that help, sleep with windows closed, use AC instead or fan (tough one as my husband is an open window person . . . ). Wash down child's room weekly, dust, wash walls, wash anything that can get dust such as all bedding, wash blinds, drapes, etc. every week. An air purifier for their room or elsewhere in the house. Keep stuffed animals to a minimum and wash the ones you keep. Change your filter on your furnace monthly. I just spent extra money on a really good filtering vacuum cleaner . . . a dyson. And my son takes an over the counter allergy pill for children, claritan, every night before bed per our doctor. All a pain but does seem to help. good luck