GroTheory 2 Pack Door Draft Stopper, Under Door Sweep for Exterior & Interior Doors, Weather Stripping Door Seal, Garage Window Blocker Noise Cold Air Insulation, Draft Guard White 2" W x 39" L

Air purifier can go wherever you want, I keep mine close to their cage since that is the smelliest. I never ended up getting a gate, I instead use these and like them better than a gate https://a.co/d/0by9bCbn I don't have carpet though… Hopefully someone else has tips for that! You can always search in the sub too.