If i buy crypto currency on an exchange with paper money. Do miners have to confirm that i now own X quantity of a BTC or whichever coin or token I may have bought, is this then confirmed by all nodes on the blockchain? Cold storage devices, any recommendations or referral to past questions or sites? thank you
Confirmation is required when cryptocoins are moved from one address to another. If you've only just bought your coins, then they're still at an address belonging to the exchange; which means the change-of-ownership is purely an entry in the exchange's private ledger and miners are not involved. The coins are only as safe as the exchange's security and solvency allows.
When you withdraw your coins to some other address (e.g. an address that has its private key in your personal wallet), then yes you do have to wait for miners to confirm the move. Once the miners have confirmed, the coins will be under the control of whomsoever controls the private key for the new address (ideally just you!).