My Coinbase BTC is not in the block-chain and I have no ETH address

After buying BTC from Coinbase they put the coins in a on-line account and gave me a public address so I understood that the
coins that they hold for me were in the public block-chain but when i checked the balance of the account using an on-line API
I discovered that the account had no transactions against it.

When taking some profit out of BTC using Coinbase the transaction completes in seconds so surly they are not waiting for the next block to be written to the block-chain

Later I also purchased ETH using Coinbase and was expecting them to give me a public ETH address that could be used to receive money from but they never did.

How would I know if they are printing these bitcoins themselves that are held in my account and would this not lead to a situation where we have a lot more
coins flying around than we think we have which is just what happened to gold/silver and resulted in market fixing.