Yes the CY36800 programmer programs the part before being soldered in circuit. The CY36800 programmer has SOIC8 socket on board.
The programmer has been designed to program the non-volatile memory of the device which is proprietary information. It cannot be used as an evaluation board to perform In-system programming.
Once programmed to your requirements using the jedec file created by CyberClocks software, the device is expected to perform as expected meeting the specifications.
I2C pin functionality can be directly used for the device with the pins provided for the same with default I2C address as 69H configured. While configuring jedec file through Cyberclocks, if PIN 3 (SDAT/FS0/VCXO/OE/PD#) and PIN 5 (CLKB/FS1/SCLK) is un-configured, then the created jedec file gets configured with I2C enable feature with 69H as the default I2C address.
Your understanding of the programmer and the CY22801 part is correct. The programmer will program the device for its power-on configuration. Changes made using I2C will last as long as the device is powered.