So, if I understand correctly, an address can receive many payments but only send one?
No. The address is "used" in the sense that it is removed from the pool and put into the wallet proper. The addresses can be used for unlimited sending and receiving.
Nearly every time you send bitcoins, you also end up creating a brand new address and sending some "random" number of bitcoins to it (back to yourself). This new address takes one address from the pool, puts it into the regular section of the wallet, and generates a new address to "top off" the pool. Wallet backups are good for only 100 "new address actions" because addresses generated after that will not be in the old wallet.
Back up your wallet. Hit "New Address" 100 times. All of those addresses are in the backup. Create one more address and you've created an address that's not in the backup -- it will be lost if you restore from the backup.
Attempting to answer my own question regarding running multiple bitcoin clients using the same wallet.dat, it seems that this would work until one of the actions listed on the backup wikipage takes place. Then the wallets would diverge and one of them would become invalid. (Or at the very least, they would contain pre-generated addresses that aren't identical, and once those addresses are used something bad would happen.)
They would diverge after a new address action, though this would only become apparent after their pools are exhausted. They'll both be able to see and spend transactions sent from and to the addresses that they share (current addresses + 100 pooled addresses). Bitcoin will synch their shared transactions to some degree by looking at the block chain, but this is not an intended use of Bitcoin and you will run into occasional errors.