Air Purifiers

So I live in a quad dorm (4 people) that has a bathroom/shower inside of it. The bathroom is right by my bed :@ so it is starting to bother me. We haven’t cleaned it since we’ve gotten here (6-7 weeks, I know, disgusting), so tomorrow I am going to force all my roommates to clean it with me. Any suggestions on what to use/how to go about this the fastest/least disgusting way possible? Last time I checked my roommates cleaning supplies (that have never been used, lol), he had Lysol toilet boil cleaner, Oust Air Sanitizer, 409 all-purpose spray, and swiffer sweeper.

I am moving into my frat house in a few days and my roomate says the room is covered in thick dust, i need to buy an air purifier but i dont know which is best, i am looking to spend under 5. Also i need to know if there is any cleaning supplies or things i can use on the carpeting which is very dusty and dirty. Also the mattress in the room is dusty. Thanks for your help.