My work is making vehicle trackers, using GSM modems.
I have never used any connections other than Rx and Tx. Various other lines are provided on the Telit modules, but I never use them.
My diagnostic route would be:-
Open a terminal emulator on the PC and select the port and baud rate.
On the emulator, make sure that local echo is turned off.
Link pins 2 - 3 on the D9 connector and check that characters are echoed.
Remove link. Connect MAX3232 and link Rx to Tx on the 5V side (pins 10 and 11 on your diagram) and check that characters are echoed.
Remove link and check that characters are not echoed.
Connect GSM modem and turn it on.
Check that Rx and Tx are high when nothing is happening.
Type AT/r ("AT" followed by a return) and check that "OK" is received.
If not, try other baud rates.