What you are describing is essentially EMV, except that your bank has gone to the trouble of picking your private key and embedding it in a card you carry around and insert into payment terminals.
Not much. The chip doesn't transmit any credit card numbers. What's really happening in an EMV transaction is the amount due is transmitted along with some identifying information from the host to the card reader. The reader then authenticates with the chip card using asymmetric cryptography. Once this authentication is done, the reader sends an amount due and the chip card checks its authorization rules, and responds with some encrypted data that represents the transaction amount and that depends on a private key embedded in the card. You could replay the transaction at the exact same time as it is happening, but you'd have to use the same amount due. And there are other identifiers for EG the terminal that you'd have to know. If you're curious, EMVco makes the specification available online in documents titled Book 1, Book 2, Book 3, and so on: https://www.emvco.com/specifications/
It's only an issue with EMV Fallback, which you'd probably not need if you have a backup card that is good. Basically if the chip or near-field antenna on your card fail, the fallback is to collect a magnetic stripe read. Properly-configured readers don't need the stripe read to complete a transaction.