2- In the yellow circle. When do I have to connect both grounds and when I should not?
Do a search for "arduino mega ugnd" for many discussions on that.
3- Why I have two ICSP in Arduino Mega (one for ATMega2560 chip and one for ATMEGA16U2-MU)? and to which one I have to use to burn the bootloader?
One is for programming and bootloading the 2560 chip and the other for the 16U2 USB to serial converter. Bootload the 2560 with the 6 pin header next to the 2560 chip. See the image.
