CrossRoads:
"Just to be clear: is it safe to power the TX module with +9V and still feed 0-5V digital signals from an Arduino pin?"
If the device supports 12V, then yes.
Ok, so it's safe, reassuring 
More power for the amplifier, separate from the input stage.
But I should use a common GND, right?
Still, does the TX module expect to receive an "amplified" data signal too? I'm sorry if I look naive but ... I'm still sending data from an Arduino pin which is calibrated on a 0-5V range, so I just want to be shure (I'm already pretty shure but I'm not an expert) that I don't need to change the signal voltage level, that if I keep it on a 0-5V range the TX module will be ok with that.