Sweep means your wallet cleans up the key you just gave it and then discards it. in other words you create a new transaction and send all the funds belonging to the key you want to sweep to a key that was already in your wallet and then throw away the key you just swept and is now empty.
Import means you just store the private key inside your wallet alongside anything else that was already in it. the key will still contain funds if it had any because you didn't touch them.
sweeping is proffered because importing a key from somewhere else may not have the same security since that key was somewhere else that may be compromised or will in the future.
So, when a DOGE wallet address is sweeped with its private key, the address is added into my wallet.
if you sweep it then it won't be added to your wallet.Quote
What if two different wallets sweeped the same address? Would both of them have access to the same address with control over its balances?
if more than one person has access to a private key then they each can spend the funds. it has nothing to do with sweeping. the transaction that is sent out first will be confirmed and the other person(s) will have no access anymore. Quote
Would the address be gone from the first wallet?
the address will remain if you have imported the key but it will be empty. 
