Old parts are great, that's why we're here
What rear mech are you running now?
What I would do personally is wait for your new wheels to arrive and try all of your existing parts prior to investing in anything new. To attempt to answer your questions:
1. No reason why not. I'm guessing that in maybe a little harder to fine tune into the right gear as they are closer together but again no reason why that cannot be done.
2. Yes and no. Cage length will depend upon your largest sprocket size, if it is below 26t then a short cage mech should be fine. You may get away with the travel of you rold mech across all ten sprockets, you may not. A 10 speed specific rear mech will ensure optimum compatibility but where's the fun in that!?
3. Double is fine.
You will need a 10 speed chain.
Hope that helps.