What if I backup my wallet from a client that is not fully synced with the blockchain?
1) Yes. You will have access to your funds. Every wallet.dat stores the "best block" (~newest block the wallet was synced with). When you start a bitcoin-core together with your old wallet.dat it will automatically rescan for wallet transaction down to the "best block" stored in the wallet.dat.
If you funds are not available, try starting bitcoin-core with the -rescan argument.

2)3) A keypool does contain pre generated keys. The keypoololdest is a timestamp of the oldest (unused) key in your keypool. If you have not generated addresses (also happened when you send funds; change address!), this timestamp can be really old and doesn't matter in your case.