12V but what Ampere?
So you want to use the 5V voltage regulator on the Arduino serve as the supply for your 5V components, and feed the 5V regulator from the 12V supply that powers your solenoid?

The problem with that plan is heat. The voltage regulator on the Arduino is a poorly heat-sinked (heat-sunk?) "linear" regulator. A linear voltage regulator is basically a solid state circuit that converts excess voltage to heat.

The higher the supply voltage you provide to the voltage regulator, the more heat it generates for a given current load.

If you feed 12V into the Arduino's voltage regulator, it needs to reduce the voltage from 12V to 5V. 12V - 5V is 7V. So it will convert 7V to heat. You're only drawing ≈220 mA. Total power that will be converted to heat is 7V*0.22A, or around 1.54 watts of power. You should budget for at least 50% more current than you think you'll need, so lets use 330 mA. 7.0*0.33 is about 2.3 watts.

From a little research I did, it sounds like a typical Arduino voltage regulator can dissipate around 1 watt of heat. You're well over that limit even at your low ≈220mA number, so you need a solution that won't overheat your Arduino's voltage regulator.

If you were to instead supply the Arduino's voltage regulator with 7.5V, it would have a lot less work to do. That's a voltage drop of 2.5V. So for a 0.33 max current draw, that would be 2.5*0.33, or about 0.83 watts. That would be under the 1 watt ceiling. However, it sounds like your project is based on a 12V supply.

You could instead use an external switching voltage regulator to reduce 12V to 5V. (search for "boost voltage regulator".) Those are much more efficient, and don't generate nearly as much heat. You could then feed regulated 5V directly into the USB input on your Arduino, as well as supplying it to your other 5V components.