Sim on port 1 is detected and I can place a call (even if the phone is still reporting "no service").
Sim on port 2 is not detected (no sim card message).

We may have to use the whole fw off the 7562 as right now, I still have the 7560 logo, then the Duos one. I tried to flash a complete stock rom through ODIN but ODIN fails when flashing system partition.

Have you already flashed a complete rom to a 7560 ?