Are my coins gone forever?

Question

I bought in 2013 and wrote down one set of numbers on a piece of paper: 33 characters, starts with 1Dao, ends on CK. I can view those numbers on blockchain dot info. It shows a transfer of 2 bitcoin, but balance zero. Am I seeing the transaction or my wallet info? However, I can’t find my wallet key. I also wrote down "MM" which I believe to be the name of the wallet?

Answer 1

The number that starts with a 1 is likely a legacy Bitcoin address. The balance shows 0 which means those bitcoins have moved. In the blockchain viewer you should see a transaction(s) that show where the bitcoins went. If you don't control the private key(s) to the final destination of those coins, you can not recover them.

Answer 2

Balance

If any blockchain explorer shows a zero balance for a Bitcoin address, it means the money sent to that address was all subsequently spent or transferred by whoever has the Bitcoin private key for that Bitcoin address.

Address

A typical early kind of Bitcoin address starts with a 1, and has a random-looking mix of about 34 numerals, uppercase and lowercase characters like this example (do not use!)

17VZNX1SN5NtKa8UQFxwQbFeFc3iqRYhen

later types of address started with "3" (same length) or "bc1" (longer)

If you enter an address into a blockchain explorer, they typically show a list of transactions involving that address.

Transaction-ID

These are usually written as 64 hexadecimal digits composed of digits 0-9 and either A-F or a-f.

If you enter a transaction ID into a blockchain explorer, they typically show the addresses used* in each of the inputs and outputs of that transaction.

Key

Public and private keys are often written as 64 hexadecimal digits composed of digits 0-9 and either A-F or a-f.

Keys are sometimes shown in Wallet Import Format (WIF). This is about 51 characters long and contains a mix of 1-9 A-Z and a-z excluding 0 O I and i.

Wallet

It isn't meaningful to ask about a wallet address because a wallet typically has many addresses and all addresses used are actually or effectively wallet addresses. For normal wallets, nothing identifies the wallet in the Bitcoin network.

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_{* More exactly: derived from relevant locking scripts. But that's a detail.}