I would put down sheet vinyl. Liam first then the vinyl, tacking down at the edges only. If it’s damaged you remove the flooring easily that way. My husband used to work for a rental home company and that’s what they did with great success.
CeCe....The first thing I would do would be to talk with landlord and make sure it is okay before doing anything,
I love rag rugs on a kitchen floor.
Id use vinyl tile or planks. If one area,gets messed up you do t have to replace the whole floor like with sheet vinyl.
Are you the renter or the landlord?
If you're the landlord, vinyl plank flooring is durable and easy to install. If you're the renter, I'd just use large rugs and ask the landlord for anything more :)