This schematic looks very bad to me.
The pointed heat pad can drawn until 600 mA and I don't think it's good idea to plug it in Arduino 5 V since the regulator can get very hot. I think the pad should be connected directly to a 5 V power supply.
About your questions:
1 - The diode seems to be a flyback but I can't understand the reason for it since the pad is a resistive not a inductive load.
2 - The MOSFET is needed because digital pin can delivery about 40 mA only,