As I understand it:
The Bitcoin address is a sha-256 hash of the public key.
The "private key" isn't really a private key but more a public/private keypair.
Yes you can easily extract a public key and bitcoin address from a private key.
Go to
www.bitaddress.org and fill in a private key under the tab wallet details... it will give you the address.
+1.
You are not keep the coin, but the private key in the wallet. The wallet software has a copy of your private key,
so it can send on behalf of you. If your computer is compromised, so does the wallet and your key.
The information on how many coin do you have is in the block chain. All the transaction history is there, and the
wallet calculate how many bitcoin you own based on that information.