It’s as reasonable a measure as any of the level of effort / output in a work, so yes it’s common in short stories and magazine pieces afaik. I’ve heard of rates anywhere from 10 cents to a dollar per word depending on genre and publication.

Generally you’ll be given a size to fit within, so it’s not like you can go Dickens on it and crank up the word count to cover your rent.

Yes, but it depends on where you submit. I've seen literary journals pay in contributor copies (aka nothing), token amounts—a few dollars, a rate per page, or a fixed rate up front. Genres like sci-fi, fantasy, horror, and crime or mystery are more likely to pay per word. Not sure why that is.