One last question. I assume that I can replace the 10K resistor with an LED and still keep the MOSFET from triggering when it shouldn't, correct?
Not sure. The 10k resistor to ground is technical not need just to turn on or off the mosfet via the Arduino output pin, rather it's to ensure that the mosfet is turned off in the event that the +12vdc is still on, but the Arduino is turned off, thus acting like a default pull-down to remove any previous gate charge voltage on the mosfet. This is a safety item to prevent the solenoid (or more likely a powerful motor) from not turning off if the Arduino was to lose power or have a reset condition, etc.
If the same power source is powering the solenoid and the Arduino (through it's external power connector) then the resistor is not really required. A LED does not conduct current unless it's forward voltage drop is exceeded, so I'm not sure it would act as a effective pull-down load or not. Worth trying as it won't damage anything if it works or not. A test would be to turn on the solenoid via the Arduino digital output pin and then remove the connection to the Arduino output pin and see if the solenoid turns off.
I guess you could have both the 10k resistor wired gate to ground and a led with it's own series current limiting resistor wired gate to ground if the first test (led only) does not work.