Can anyone explain what would happen if I created a wallet in 2010 and someone sent coins to the wallet address, gave me a tx number that I stuck in and hit receive, shut the wallet down and left Bitcoin demon running ( It probably didn't sync with the network) and went to work. When I got home the power had been off and the computer was off, I never had it back online since then. I looked on the block chain at old addresses and don't see that amount around the time it happened.
If someone sent bitcoins to one of your addresses and the transaction was mined onto the blockchain, the bitcoins are at your disposal if you still have the private key (or the mnemonic phrase). Whether your node happened to be online and synced doesn't matter. But if a reliable blockchain explorer shows an empty balance for the address, then the balance for that address is empty.