Correct. The fare class does not change if you change flights.

The value of your ticket is what you paid for it, not what it's priced at now. If you want to sit in Mint, you will either pay the fare difference between what you originally paid and the current Mint fare, or you can buy up at the gate for a flat fee if there are any seats left by then. You will not be have the option to pay for the difference after the flight goes to gate control.