Your wallet doesn't actually store any coins. What is stored is identification numbers (addresses) that allow the wallet program to look in the blockchain to find out what value is associated with those addresses. All the records of how much value is associated with each address is actually stored in the publicly shared blockchain.
So to answer some of your questions:
What would happen if I had 2 instances of the wallet open at the same time in 2 different locations, and someone sends me 1 bitcoin. Would both wallets on the network receive the coin?
Yes, both instances of the wallet would look at the blockchain and see the transaction there, so they would both show the additional coin.
I just had a quick question on how the wallet operates. I'm about to take a trip for a week, and will be taking my wallet with me. My question is how exactly does a wallet operate? For intense: I backup my wallet as I normally do. I take my trip. During my trip I add a few funds to it. I come home, and restore the updated wallet.
But what happens if I don't restore the wallet? In other words, I load up my week old wallet, and try to receive or send some coin?
This depends on how many new addresses were generated by your wallet prior to loading up the old copy. By default the client pre-generates 100 addresses that it keeps hidden from you. When it needs a new address it grabs one from this pre-generated list and generates a new one to add to the list.
So if you use less than 100 new addresses, then all the addresses you used will still be available in the old copy of the wallet. All the coins will be seen there and accessible.
If you use more than 100 addresses, then any coins that were sent to addresses after the first 100 will not be visible in the old wallet since it won't have the private keys to look them up in the blockchain and they will therefore not be accessible.
I don't fully understand how this all works. Say I had 5 bitcoins to start with. Take my trip. Send those 5 coins away. Come home and load up the older copy still on my main PC. Will those 5 coins still show up in the older wallet?
No. The older copy will use the addresses to look in the blockchain and see that those coins are gone.
What happens if I try to send them?
It will be seen as a double-spend attempt and the peers will not forward the transaction and the miners won't include it in any blocks.
What happens if I add coins to the old wallet even after sending the coins away from the newer copy?
Coins aren't added to wallets, they are added to addresses. As long as the new wallet has the same addresses as the old one, any coins sent to addresses in the old wallet will be visible in the new wallet as well.