One private key in different locations?
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.