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Mycelium for Android, and Breadwallet for iOS are good choices.
"Sweeping" is the process of sending all your funds available to another address. Unlike you importing an old address to a new wallet, sweeping requires a transaction between two addresses, since you are sending coins over the blockchain.
Is the same as you going to the "send" tab of your wallet and sending every Bitcoin you have to a new address. There is nothing special about this process.
When Should You Sweep a Private Key?
You should sweep a private key if someone else has or will ever have access to the private key (like if someone gave you a paper wallet or the private key was published online). This will prevent a thief from ever being able to spend the bitcoins associated with it, even if they have or ever get access to it. If the private key (or paper wallet) will be used again in the future then you can keep it and sweep its contents again once more bitcoins are sent to it.
If youre still in doubt of whether to import or sweep a private key, you should probably sweep it. It solves many of the issues that can arise from doing an import.