grrigore:
How do I have to connect the module to an Arduino Mega 2560? Do I have to upload some code on the board in order to get access from the phone?
No
Edit: How should I set my bluetooth in AT mode if I want it to send a *.txt file from Arduino to an Android app?
You don't need any AT commands to do that. Indeed to use bluetooth, you may never need any AT commands, ever.
Bluetooth RX --> Arduino RX (pin0)
Bluetooth TX --> Arduino TX (pin1)
Wrong. Transmitters transmit to receivers, and receivers receive from transmitters
Tx>Rx Rx<Tx
Note also that it is a good idea to use a 1k/2k divider in Arduino Tx line. Also, since you are using a Mega, you will be better off using Serial1 pins 18,19.