carbon filter push air through air cool hoods?

yes no

sure why not, air can run either direction

if you push air, it will put pressure on the seals and leak air. if you pull air, it creates a suction,lessening leaks.

Filter the air prior to it entering the hoods or you'll be cleaning out the insides of those hoods a lot.

Also, shouldn't the air going through the hoods be on a different circuit then the air in the room?

OP, why don't you use more words and some actual detail and let us know what you are asking, what you are planning.

filter > ducting > hood > ducting > fan > ducting > outside

for a single exhaust setup this is how i would do it

I go Filter>Fan~~>Ducting>hood>ducting~~>exhaust outside

This topic has been addressed so many times already