Presumably there is a regulator between the 12V supply and the arduino. As long at the 12V supply remains high enough for the regulator to function, it will keep the voltage to the arduino constant.
However, regulators aren't perfect. In particular, they can only compensate for input power variations up to some frequency. Noise above that will be passed to the output. If you think the input power might be noisy enough so that the regulator can't block it all, put a small filter on the input of the regulator. Most regulators will work OK up to a few 10s of KHz. You didn't say how much current the arduino draws, but you have plenty of voltage headroom. A 10 Ohm resistor will only drop 1V at 100mA, which still leaves plenty for the regulator to work with. That followed by a 10uF ceramic cap to ground right in front of the regulator will form a low pass filter with 1.6KHz rolloff. That should be good enough to keep the input of the regulator down to frequencies it can deal with.
Of course the best way to handle noise is to not make it in the first place. You absolutely need a diode or some kind of snubber to catch the inductive kickback when the solenoid is turned off. The main reason is to keep from frying whatever is switching the solenoid, but a secondary effect will be to decrease overall noise.
FET Type: I'm not sure what the difference is between N and P channel
The internal construction of a mosfet is different and you need different voltage levels to switch it on. Higher than source for N channel and lower than source for P channel. As you will be switching 25V load from a 5V microcontroller, choose an N channel logic level mosfet.
Drain to Source Voltate (Vdss): I'm assuming this is the max voltage it can handle going through it, so I should be finding a MOSFET that will support 25 V+?
It's the maximum voltage whitch the mosfet can withstand without letting the current to run through it.
By the rule of thumb you should double the rating to get a reliably working system. So, look for a mosfet with Vds in the range of 50V-60V. It would be OK to use a 25V mosfet but you usually don't want to operate near maximum limited values.
Current - Continuous Drain (Id): Assuming this is the max amperage going through it, so looking for one with 12.5 A+
Again - double it.
Vgs(th) (Max): I think this has something to do with the activation voltage applied to the gate that will make it activate, so I need one with less than 5 V?
Yes, mosfet dissipates least power when it's either fully on or off. Look at the graphs in the datasheet that specify Rdson depending on Vg - you want Rdson as small as possible, so you want to drive the gate above the Vgth. But note, that there is a maximum value that can be safely applied to a gate - Vgsmax. You should be safe driving it with a microcontroller, just a point to note.
Power - Max: Assuming this is the max power it can handle. I've calculated the power the solenoid would need as P = V*I = 25 V * 12.5 A = 312.5 W, so I need a MOSFET that can handle more than 312.5 W?
No, power dissipated by a mosfet would be I*I*Rdson - that's why you want as little Rdson as possible.
I don't know what Rds On (Max), Gate Charge (Qg), or Input Capacitance (Ciss) mean. Are they important for my uses?
When a mosfet is on, it's not an ideal conductor with no resistance. Rdson is the resistance of the mosfet and is dependent on different factors, datasheets usually give graphs how Rdson changes with different parameters.
You don't have to deal with gate charge and input capacitance in you application as fast (submilisecond) switching is not required. A mosfet gate presents itself as a capacitor to a driving circuitry and as it takes time for a capacitor to charge, it takes time for a mosfet to turn on that's why in high speed applications special mosfet driver ics are used that force high currents into gate to charge this capacitance as quickly as possible.
You can find cheaper mosfets with lower Rdson, just use the parametric search on digikey. Pay attention to the graph that displays Rdson against Vgth - sometimes manufacturers claim 4V Vgth and 4mOhm Rdsn, but when you look at the graph you see, that at 4V it's 20mOhm and you need to get to 9V to get the advertised 4mOhm Rdson.
Best Answer
Those are common symptoms of layout problems, or possibly power supply issues if your 5V supply is derived from your 25V (like through a buck regulator). You want to make sure the ground paths are separate for your 5V and 25V sides, then meet at exactly one point as close as possible to the supplies. If that's not sufficient, the next step would be completely isolating the circuits with an optoisolator.