Is this specific board "ESP32-S2-DevKitC-1" compaitble with IoT Core, FreeRTOS Quick Connect and setup, etc?

Hi. As you have an ESP32-S2, I think you'd be better of with this (similar) guide: https://docs.aws.amazon.com/freertos/latest/userguide/getting_started_esp32-s2.html

That targets the ESP32-S2-Saola-1 board which has a different MCU than yours. Your specific board is not supported out of the box by the reference implementations in the amazon-freertos repo. Besides the AWS documentation pages, you can see the supported boards here: https://github.com/aws/amazon-freertos#freertos-qualified-boards

And here are the CMake recipes for the supported ESP32 boards:

• https://github.com/aws/amazon-freertos/blob/main/vendors/espressif/boards/esp32s2/esp32s2_saola_1.cmake
• https://github.com/aws/amazon-freertos/blob/main/vendors/espressif/boards/esp32/esp32_devkitc.cmake
• https://github.com/aws/amazon-freertos/blob/main/vendors/espressif/boards/esp32/esp32_plus_ecc608a_devkitc.cmake
• https://github.com/aws/amazon-freertos/blob/main/vendors/espressif/boards/esp32/esp32_wrover_kit.cmake

Your board has the ESP32-S2-SOLO MCU, which is a different MCU than the Saola. More on the ESP32-S2 boards here: https://docs.espressif.com/projects/esp-idf/en/latest/esp32s2/get-started/index.html#hardware

I suspect it's not a lot of work to adjust the Saola build to suit, but I've not done it before (and may be underestimating!). I think, at minimum, you would need to make an esp32s2_devkitc.make that was similar to esp32s2_saola_1.cmake. Then follow the esp32-s2 instructions, adjusting as appropriate.

Does my specific devboard support quick connect and all the benefits listed here?

No, that MCU is quite different. It has a secure element for example.