What if I backup my wallet from a client that is not fully synced with the blockchain? I've just backed up an encrypted wallet, but the client (Bitcoin-Qt) is 26 weeks behind. 

During these last 26 weeks this wallet has only received BTC (roughly one transaction per day), no payments were made from it. 

I have a few questions:

(1) If I install a new client on a new machine, import wallet from this backup, and then let the new client fully sync with the blockchain, will I have access to the BTC received during the last 26 weeks, or these funds (or some them) will have been lost?

(2) I have run the keypoololdest, which returns May 10th 2014; I assume this is the date of the oldest unused key in the pool. The last outbound payment from this wallet was in May 2014, about a week after the keypoololdtest date.

The date which is 26 weeks behind today is Jan 29th 2015, which is 264 days after the keypoololdtest date. 

Could someone please shed some light on what's going on here?

(3) Are keypool keys only used on outbound transactions (to receive change), or each inbound payment also uses up a key?

Thanks in advance!
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.