No. Instantaneous current will vary as a function of coil inductance, coil resistance and the equivalent series resistance of the capacitor. If the solenoid has a moving ferromagnetic core then the coil inductance will vary with time.
How would that prevent the faulty check control unit signalling low fluid level? Would removing the relevant bulb be enough? Or would there still be an error flagged and stored by the unit?
The power dissipated in that piece of wire will be 2A x 2A x 12mΩ = 48mW. I doubt it will get noticeably warm to the touch.
If you're trying to make a heater, use resistance wire (e.g. nichrome) instead of copper
50cm of 1mm diameter copper wire has a resistance of about 12mΩ. So a 12V source will try to drive a current of about 12V/12mΩ = 1000A through it. That would vapourise the wire virtually instantly.
You haven't said whether 2A is the rated current of the supply or whether it is a controlled...
It looks to me as though the servo has a linear feedback potentiometer for position control. Won't you need to use this in your project?
If so, the circuit required will be more complex than an H-bridge.
Unless it's a camera artifact there seem to be soot deposits on two of the fan blades, close to the fan centre, suggesting at least one of the transistors in that area emitted the magic smoke.