The circuit is fine in theory.
Improvement in practice is required.
Adding a gate-source zener diode of say 12V (> Vgate_drive) is a very good idea indeed in all circuits with inductive load. This stops the gate being driven destructively high by "Miller capacitance" coupling to the drain during unexpected or extreme variations in drain voltage.
Mount the zener close to the MOSFET.
Connect Anode to source and Cathode to gate so that the zener does not usually conduct.
The 10k gate drive resistor (as shown) is large and will cause slow turn off and on and more power dissipation in the MOSFET. This is probably not a problem here.
The chosen MOSFET is very marginal in this application.
Far far far better MOSFETs available ex stock at Digikey include:
For 26c/10 Digikey IRLML6346 SOT23 pkg, 30V, 3.4A, 0.06 Ohm, Vgsth = 1.1V = gate threshold Voltage..
NDT3055 48c/10 TO251 leaded 60V, 12A, 0.1 Ohm, Vgsth = 2V
RFD14N05 71c/10 TO220 50V, 14A, 0.1 Ohm, 2V Vgsth.
ADDED
SUITABLE MOSFETS FOR 3V GATE DRIVE:
System just trashed my longer answer :-(.
So - MOSFET MUST have Vth (threshold voltage) of no more than 2V to work properly with 3V3 supply controllers.
None of the suggested FETS meet this requirement.
They may work after a fashion on the present load but are underdriven and overly lossy and the solution does not extend well to larger loads.
It seems that IRF FETS in size range concerned that have Vth (of Vgsth) <= 2 volts ALL have 4 digit numerical codes starting with 7 except IRF3708.
OK FETs include IRFxxxx where xxxx =
3708 6607 7201 6321 7326 7342 7353 7403 7406 7416 7455 7463 7468 7470
There will be others but all the ones suggested seem to have Vth = 4V or 5V and are marginal or worse in this application.
Vgsth or Vth needs to be at least one Volt less and ideally several volts less than actual gate drive voltage.
First part of your question can be answered experimentally. Charge those capacitors to 15V, disconnect them from the 15V source, and apply them to the solenoid. If the solenoid operates acceptably then the capacitors have enough capacitance and you're good to go from that perspective. Otherwise you'll need more capacitance (or more voltage!).
Secondly, you may need some way to isolate the drain of the solenoid from the boost regulator output to prevent it from dragging down the input voltage. The easiest way to do that is to use a resistor that is high enough value to prevent overloading the source and low enough to allow the capacitors to charge in a reasonable length of time. Suppose you allow a 25mA charge current, which might be 75mA at the batteries. That would be a 600 ohm resistor, so let's try 680 as a standard value. Power dissipation should not be much but if you use a 1/2W resistor it can't get too hot even if you leave it on.
The time constant of 680\$\Omega\$ and 4400\$\mu\$F is about 3 seconds, so after about 10 seconds or so the capacitors will be fully charged.
Best Answer
Often a punch-through failure on the gate due to ESD can cause the gate to fail with some voltage on it leaking through from the body. If this is ESD damage, then handling the FET according to safe ESD practices will prevent future failures.
Keep the FETs in ESD safe packaging until needed. Use a wrist strap that's correctly grounded when handling the FETs. Do assembly on a bench with a grounded ESD mat.
Also, some newer FETs have gates that are only rated to 8V. Make sure your FET gate is rated for 12V operation.