1) I have the bitcoin client downloaded on 2 different computers. How do I, if I can, use the same account on both of them? For the record, I have only used one of the clients so far (to obtain, but not yet spend, coins), but prefer using the other one. If I have to ditch one, can I transfer my coins to the other?
It's not recommended to try to use the same wallet on more than one machine. Consider using remote login or an eWallet service.
To move permanently to a new machine, you need to locate your wallet.dat file in Bitcoin's application data directory and move it to the new machine. Make sure that Bitcoin in the old machine doesn't try using the file after you've copied it. The first time you run Bitcoin on the new machine with the copied wallet, use the -rescan parameter.
2) Suppose I need to do an OS recovery on my computer, and have to re-install the bitcoin client. How do I recover the public and private keys, or are my coins lost? For the record, I have downloaded the truecrypt program, so I can make a backup of my wallet on a thumb-drive.
After you reinstall Bitcoin, delete the wallet.dat file it has created and put instead the file in your backup. Use -rescan the first time.
3) How can I access my private key? The only key I see in the client is the one in the receive coins page, which should be the public key, as private keys usually begin with a 5 (according to the wiki).
This isn't something the Bitcoin client supports. You can use pywallet