Thanks very much for all the suggestions.
Eventually I succeeded obtaining DIO0 interrupt indication TxDone, although, for this to work I just put a 10 ms delay between SX1276SetOpMode( RF_OPMODE_SLEEP ); and the registers' initialization loop in SX1276Init function.
However, the problem now is with the RX, i.e. its absence. The RX part ends with RxTimeout. So the DIO0 does not go high during the RX part. I have two identical devices both of which shows TxDone but neither of them reaches RxDone.
I understand that this information is too abstract to suggest anything specific, but will be very grateful for any hint.
Thanks a lot in advance.