One private key in different locations? Can I still use my key pair to receive and send bitcoins, even if I imported the private key in another location? I wondered about this, when I planned to import a backup of my online wallet in a pc-client, but still wanted to use my bitcoin in the online account.

Or is only the latest location where I imported my private key able to operate with the bitcoin? (I'm still trying to fully understand the bitcoin universe)
Bitcoin works in a way that you don't actually own any coins, but instead you have the key to unlock them and spend them - so that you can send them to someone else who can then unlock them and spend them.

Using that logic, anybody that holds your private key can unlock and spend the coins from any computer anywhere in the world with an internet connection.

In terms of wallet implementations, it's a bit tricker to use the same wallet on different computers as there will be synching issues between the various wallets. The "change" from a transaction will go into another address in your wallet, which might not be present in your other wallet. Please read Wallet on multiple computers

Again, this depends on your wallet implementation. If you're using just the raw private key to make transactions, it shouldn't matter how many copies there are or where you broadcast the transaction from.