Bitcoin wallet offline
Franz-Josef is right, a cold storage wallet does not mean you are taking the bitcoins offline.
To clarify, bitcoin is a public ledger. Each address ("public key") is visible to anyone, and anyone can see the amount of bitcoins stored in that address (just use any bitcoin explorer, e.g., ). If any address receives bitcoins, the balance of that address will be updated instantly.
The private key of an individual address is only necessary in order to initiate transactions (= make/send payments) from that address. In a cold storage wallet, you are taking this key to an offline location, e.g., a piece of paper, a flash drive, or a computer not connected to the internet.
So to answer your initial question: You will receive the 10 satoshis at the time somebody sends it to you, and you can see the updated balance on the public ledger instantly. You will however only be able to make payments from that address once you take the private key back online.
- When large bitcoin companies such as exchanges manage bitcoins, they typically have multiple addresses in cold storage, and send newly received bitcoins to the cold storage addresses all the time (which resembles your question). When they later need to make bitcoin payouts which exceed the amounts they have available in online wallets, they just take one of the cold storage addresses of appropriate size back online.
- The (obviously flawed) email analogy for bitcoin holds here: If you have a gmail account, you will still receive the email, even if you have your password only written on paper (cold storage). Once you take your password online, you can access it - but the email had arrived before.
- When people lose or unintentionally destroy their private keys, the bitcoins are "technically" not gone - the corresponding addresses are still in the ledger for everyone to see, and are still able to receive coins. It's just that nobody can access the addresses anymore to spend (which, to be fair, has the same effect as the coins being lost).