It's theoretically possible. However it wouldn't be an elevator being that up and down are relative terms, and first your would have to bridge the gap in between two stars which would take many lifetimes even at light speed. And the maintenance of such a thing would make it a waste of money and time. So no it won't happen.
I highly doubt it. It would be increedibly hard to built, and I personally think it would explode once it was out of an area with graviy. (I.e space).
There are so many reasons why this would be impossible I can't even think of how to begin...
It is practically impossible. It needs lots of capital and maintenance, and without gravity supporting it, the elevator would spin like crazy in space.