Yes, you should be able to do that as long as no two running instances try to access the same copy at any one time (which is not what you describe). I'm not 100% sure if there are issues if you copy the format between operating systems and processor architectures, but at least if you use a comparable system on each computer, it should work. Make sure to delete the wallet file, though, unless you explicitly want that each computer runs with the same wallet. (This may produce conflicts when people try to spend coins at the same time.)
Yes, of course every machine should have its own wallet.
Do you think copying the "blocks" subdirectory to all machines is enough?