Quote from: cosmarchy on July 26, 2013, 04:58:50 PM

so I guess you have to be online for the software to check it is creating a unique wallet address no one else has then I you just go online periodically just to 'sync' up with the network and see your balance. Is that correct?

You only have to be online to check your balance (you can do that with just the address, no need for the private key, e.g. here: http://blockchain.info/) or to spend Bitcoins (private key needed).
You don't have to be online to create a new address (it's even more secure to create it offline), the chance of an address collision is extremely slim, even when you create millions after millions of addresses.

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What happens if you don't go online for a month, for example? If your balance is floating around in cyberpace, is there a timelimit where it can just get lost, swamped in the ever changing, growing data?

No time limit at all, the blockchain stores all of them.

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The second question is how do you back up the private key? I see there is an option to back up the wallet but I cannot see an option for the private key.

There are different possible ways:
Either make a backup of the wallet.dat file (C:\Users\[Username]\AppData\Roaming\Bitcoin or use file->save wallet)
Use a command in the Help->Debug->Console to obtain the private keys
Create a new address (e.g. at https://www.bitaddress.org), write down the private key somewhere and import it into bitcoin-qt to use it (also with a command)

Important to note about bitcoin-qt:
It will send change (e.g. you got a transaction of 0.8 Btc and want to send 0.5 now, you will send all 0.8 and get 0.3 change back) to a new hidden address, after about 100 transactions with change it will create new ones, so they aren't in your backup anymore.
I would suggest using the addon Coin Control with bitcoin-qt, it makes this way clearer:
https://bitcointalk.org/index.php?topic=144331.0
Also remember that whoever has access to your wallet.dat has access to the coins.

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I assume the wallet balance is the total amount held within each of the wallet addresses

correct.

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How can you delete addresses?

You can delete the wallet.dat, the next start it will create new addresses.

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Once I'm synced on the network, can I 'export' the blockchain to my other computer so I don't have to download it again?

yes
(Don't copy the wallet.dat and use it on both computers, it works, but it will most likely get messy)