arduino circuit review/ safety review
The "Arduino" part of your schematic looks OK. 9 VDC to VIN on the Uno is OK. Connecting the MOSFET directly to the Arduino is OK.

Personally, I would connect a 1000 ohm resistor in series between the Arduino pin and the gate of the MOSFET. If you like to live dangerously like I do, and change wiring connections with power supplied to your Arduino, then a 1k resistor will limit the maximum current flow to around 5 mA. This is not going to "fry" anything if you inadvertently connect an output to power or ground.

The question about the L7805CV is off-topic, but a quick look at the datasheet suggests an input and output capacitor should be used.

I’m sorry to be the bearer of bad news, but the IRF540N is a poor choice as a MOSFET to use with a micro-controller. While it’s true that you are only drawing 60 mA, the VGS specified for this MOSFET is at 10 VDC, not the 5 VDC supplied by the Uno. A "logic level" MOSFET would be better suited for your application.

For more information regarding MOSFETs, please read: How to control a 12V solenoid valve with a mosfet?