i don't know about all that...
it seems like you are over complicating a circuit with unusual parts that "simply" needs to have an equally complicated mosfet driven solution.

I'll tell you why... A solid state relay is basically a low impedance mosfet switcher on the output side anyways. by using it in this manner you suggest, you may as well go straight to a MOSFET switcher using, say an IRL540 to drive your solenoid.
i would suggest using a 4N35 optocoupler to protect your jamma signal from the button. your board WILL NOT like 24 volts run into it if anything fails. it's output can be tied to your circuit trigger easily.
you can easily build an adjustable 555 timer solution:

you'd just have to modify this circuit to "one shot" your timer circuit instead of being astable.
i'd recommend building a small circuit to control an led or small lamp first. then work your way up.