I'm fairly certain that they do not
Though, in her case, she never was alive so she doesn't have a place to necessarily call "home". So, I guess in her case having a place where she can go unwind or a thing to call her own might be good. That sheet is definitely made out of the same stuff her clothing is made out of therefore it doesn't feel bad to her and she can actually interact with with it. That sheet and her clothing is basically half ghost/half corporeal.
Regardless of ghostly logistics and ectoplasmic bedding, every R.I.N.G.S. gets an apartment/dorm and this applied to Lydia as well. Since she doesn't have any "things" the only thing in her room is a bed with that sheed
Okay, I way over thought that but whatever
Morton is moé. At least as moé as a horned and disembodied skull can be
I hope you like her