Yes on the Sew Your Own thing. I made a "fitted sheet" for the crib mattress and it works to cover a dog bed, too. I need to build more of those because it is easier than washing the covers. It's an easy project, too. You could just do a partial casing on each end & run elastic through that.
The thing with rugs is I need to use the ones without backing because, in my experience, the non-slip backing comes off in the wash just enough to louse up the machine with each washing.