Combining Air Conditioner with Air Purifier
My wife is asthmatic and has strong allergic reactions to dust, mold and dust mites, so we deal with this very condition.

There are some 'portable' air conditioners that draw air from inside the room and cool it, while also cooling the room by blowing the heated air outside. These come with rudimentary filters, but also have the added benefit of creating a slightly negative pressure environment which can help remove allergens.

Additionally, because they are actively condensing some moisture from the room, it will additionally remove some pollutants. I know this is true anecdotally as I have to infrequently clean the condensing coils and filter regularly and they are often dark and extremely soiled.

This unit sits on the floor, and uses a plastic shroud (that must be fitted) to seal the exhaust out the window.

I believe this fits most of your requirements. Although I did not notice it at the time we purchased ours, I believe there are some models available with HEPA type filters also.