Gentlemen,
Thank you all for your support and effort.
I just made conductivity test, the result is (IMO) shocking.
ICSP Pin 3 (GND) on the adapter board has connection to the following pins (with reference to the PIC18F2550 pin-out (no PIC in the socket): 5, 7, 8, , 12, 18, 19, 24 and 26.
I am convinced that the board is faulty, maybe several shorts in the multilayer pcb.
After this I am not surprised about the many negative reviews of the item on Amazon.
I am undesided,whether I should ask for replacement or try to find a different product.
BTW, with the PICkit3 in stand alone mode (adapter not connected) the test
"Tools" ---> "Troubleshoot..." was ok.