Which are the minumum requirements to work with the caravan just to run my own firmware to test my circuits through analog_gpio? I mean, which voltages and pins and clock I will need to maintain once firmware was flashed using caravel hat board. I will need to do another board for futher tests. My plan is to flash the firmware using efabless boards and with "good" chips, then dessolder the chip and solder in the other test board.

You will need to power the SoC and provide the SPI flash to configure the GPIO pins for your project, implying that you also need to provide the clock signal. If you connected only the 11 analog pins across the top, and are not using any of the other 27 GPIO pins, then you should be able to just power up and test without running the SoC. I would need to know what I/Os your project is using and what you intend for your firmware program to be doing, to be more specific.

The analog pins across the top are ok. But I have also circuits using gpio_analog from 0 to 16 and my question was more about to test them with the firmware that I am testing using caravel hat (without the hat).

You should be able to use the same firmware under the same conditions. There was a time when I thought that the hold violations were behaving differently when the board was mounted on the Nucleo vs. when it wasn't, but now that I have a better understanding of the errors, that doesn't seem to be the case. However, you do want to be careful to keep the operating voltage of the vccd 1.8V domain very close to what it was when the I/O was calibrated.