So I've read that a wallet is basically a public address key paired with a private key.
But is it more than that? Does the physical wallet file contain transaction and balance information as well or does that get pulled from the cloud?
For example, let's assume I have just the public and private keys written down on a piece of paper...
If I import those keys into a wallet client, will the wallet client download my balance and transaction history associated with my address from the cloud?