[LT_2022]

I'd see a doctor for your coughing / asthma-like issues first. They may be unrelated to the rats.

There are things you can to to minimize allergens. Use fleece liners instead of bedding. Some bedding can be very dusty. Run an air purifier in the bedroom. Don't let the rats on the bed. Sleeping with bedding covered in rat pee and poop can't be sanitary or healthy. Vaccuum the room daily to remove any poop and dirt on the floor. Move the rat cage to a different room in the house. Etc.

 

[LT_2022]

I have cats, too. My rat cage is in the living area. My cats ignore the rat. Your cats, however, may react differently to a rat.

You can either don't do free roam playtime (I don't) or only allow the rats to play in the bathtub, supervised of course and with the door closed.