Air Purifiers

I am trying to fight against allergens in my house, as my boyfriend is allergic to my cat.

I was considering buying a HEPA air purifier for the bedroom and a larger one for the kitchen/dining room/ living room area of my house. I’m also planning on replacing the furnace soon and could have a HEPA filter installed on the furnace then as well.

Would one of these be better than the other in term of cutting down allergens in the air? Financially, the individual units seem cheaper, but I feel like the furnace filter might be more effective as it would help the whole house.
Good individual air purifiers cost anywhere from 0-0 and I would have to purchase multiple. Getting a top-of the line furnace would cost even more. "Doing both" isn’t really an option.

I live in an apartment here and I am getting sick of all this crap from other people's cooking. It's an older building and they won't look into it.
I read HEPA is a better purifier but I don't know if it would stop smells from coming in my front door… it is shut so don't give me stupid stuff about shut your door, just too old of a building (renovation they are doing doesn't help).
Smells only hit me in the living room/dining room… about 350 sq ft I need to protect from smells.
Will a HEPA do this?