The second construction is used very rarely, if ever, today. The only time I could imagine someone saying something like this with a verb other than do is with have e.g. "Have you enough to drink?" and even then I would avoid it (unless it's part of a perfect verb). You're always safe with a form of do.
As far as I know "Have you a computer?" is a typical one expression from British English. In GB they use verb "have" without the auxiliary "do" in questions and in negative statements.
That sounds very natural to me... the last one was one of my Mum's favourite expressions.Yes they do it all the time,
Have you any brothers?
Have you an idea?
I haven't a clue...