1. Even though Coinbase indicates that I own the bitcoin, the blockchain contains no record of my purchase, or of any transfer of bitcoin to me. Correct?
Correct.
There is no transaction on the blockchain to be shown.
Coinbase displays a number as your balance and promise that you can withdraw that. This promise can be broken at any time.
2. If there is no record in the blockchain, and if Coinbase disappears tomorrow, then my ownership of bitcoin also disappears. Also correct?
Correct.
As long as your fund is on coinbase, they have full control over that.
If they disappear for any reason, you will lose your money.
3. I guess you are telling me that, as a general rule, whenever I purchase bitcoin, I should promptly transfer the bitcoin to my cold wallet. Again correct?
It's recommended to move your bitcoin to your own wallet. In this way, you can have full control over your fund.
If your wallet is a cold wallet, you can be more secure.
4. And lastly, when I do transfer the bitcoin to my cold wallet, I then will see a TXID showing a transfer to bitcoin to me? But what will the other side of the transaction look like (ie. from whom will the bitcoin be transferred)?
Your wallet should show a same transaction.