Can addresses be split up into separate wallets, or do they share the same private key(s)?
yes, but that's just how bitcoin-qt.exe works, it does not have to work like that / someone could write a different program that say ... sends change to a specified change address instead of making a new one
As HumpNRun said, sanga's description is a pretty good explanation of how the Bitcoin-Qt wallet works. Other wallets (such as Electrum, MultiBit, Armory, and https://blockchain.info/wallet ) handle it differently.
The other thing to be aware of is that each output received is tracked and spent separately. If you receive a 5 BTC transaction to an address and then later you receive a 3 BTC transaction to the same address, you don't have 8 BTC combined at that address. You have a 5 BTC output and a 3 BTC output that can each be spent separately.
Also any output that is spent MUST be completely spent.
So given my example of a 5 BTC and 3 BTC outputs, If you were to create a transaction to send 2 BTC to someone, the wallet would choose one of the two outputs (lets say it chooses the 3 BTC output for this example) to use as an input to the transaction you are creating. Then it would create 2 outputs for your transaction. One would be a 2 BTC output to the address you are sending to, the other would be the change (in this case 1 BTC) that would be sent to an address in your wallet (in the case of Bitcoin-Qt, this would be a new address that the wallet wouldn't bother telling you about).
So, you'd still have 5 BTC associated with the original address, and 1 BTC associated with the new change address.
If you then created a transaction to send 5.5 BTC to someone, the wallet would spend both of the unspent outputs, including them as inputs to the transaction you are creating (for a total of 6 BTC), then create 2 outputs. One would be for 5.5 BTC to the address you are sending to, and the other would be for the remaining 0.5 BTC that would be sent back to an address in your wallet (in the case of Bitcoin-Qt, again it would create a new address that it wouldn't bother telling you about.