With the secondary open circuit, only the primary winding has effects on results and therefore the current that flows is the magnetizing current. The magnetizing current produces losses and these are called iron losses and they are partly due to eddy currents circulating in the the iron laminates that make the core. Iron isn't the best conductor of electricity and so it's resistance dissipates power due to these eddy currents.
Hysteresis loss is also measured by this test and, strictly speaking the total "loss" with the secondary open circuit is eddy current loss plus hysteresis loss (Remanence effects in the iron material leading to energy loss every cycle of ac).
Shorting the secondary tests copper losses and this is usually done at a much lower applied voltage to the primary. Because of this, iron losses are negligible and can be ignored. In this test you are measuring the I^2*R losses of primary and secondary together. If you know the turns ratio you can make reasonable assumptions about the distribution of loss in primary and secondary.
Below is the equivalent circuit of a transformer with the secondary (copper losses) referred back into the primary. This referral is done by multiplying them with turns ratio squared: -
Note component Xm - this is the reactance of primary inductance of the transformer. Xp and Xs are leakage reactances (turns of copper that don't couple magnetically).
Measurements
Watt-meter measurements will indicate the losses on both tests.
Ammeter tests will tell you the current in the primary and/or the secondary - this can also help you gain knowledge about the turns ratio. Ammeter tests also tell you that you are at the limit of current for testing with the secondary shorted i.e. at the manufacturer's recommended limit. Don't go higher is the rule.
Voltmeter tests only apply to primary testing when the secondary is shorted and apply to both primary and secondary when the secondary is open circuit.
If you don't have a watt-meter, voltmeter and ammeter tests plus a little bit of common sense and knowledge can help you calculate powers that would be indicated by the watt-meter.
This answer applies to regular power transformers - it isn't intended to cover high frequency transformers although the principles are the same BUT dielectric loss and resonance effects can play a big part.
This strongly smells of the solenoid return currents and inductive kickback paths not being handled properly. There are large and fast voltage spikes at the solenoids. Sometimes one of these couples enough to the microcontroller to confuse its internal logic. The reset mechanism is being tripped, but not by the external MCLR pin.
Absolutely the first thing you must do is ADD A BYPASS CAP across the micro's power and ground pins! Put a 1 µF ceramic cap physically as close as possible between the power and ground pins. This is exactly the kind of symptom a lack of bypass cap would cause.
Other than that, there are two remaining obvious suspects: poorly designed power and return current paths, and poorly handled inductive kickbacks.
Your schematic doesn't give us any idea of the physical layout of the power and return currents to the solenoids. The current loop of power supply to solenoid and back to power supply should have as little in common as possible with the microcontroller power loop. For example, if the two share a significant section of a ground wire, then the high solenoid currents in that ground wire could cause a ground bounce for the micro.
Ideally, there are separate power and ground feeds to the solenoids and the digital circuitry, with these connected at only one place close to the power supply. Then of course there needs to be proper bypassing of the power at each point of use on the digital side.
You do have a diode that is supposed to catch the inductive kickback, but you haven't shown any specs. No, a 1N400x is not appropriate here. I'd rather see a Schottky diode, due to their very fast response times.
Placement of the diode is also important. It is good to have some protection at your driver circuit in case stuff happens, but to really deal with inductive kickback it should be shunted as close to the source as possible. You want to contain the nasty current in as small and local a loop as possible. Small minimizes its radiation and capacitive coupling to elsewhere. Local keeps it from causing ground bounces and the like to other parts of the circuit.
As a experiment, try adding Schottky diodes in reverse across each solenoid right at the solenoid. Perhaps you can't put them there in final production, but do the experiment anyway to see if things change.
I suspect by observing proper hygiene, things will work a lot better. After you fix this mess, reflect on all the times you were told to use bypass caps, carefully place return current paths, keep the loops small, etc, and you thought "bypass schmypass, blah, blah". Now you know why it matters. Yes, you can get away without this sometimes, but sooner or later it will catch up with you. It just did.
Best Answer
MOSFETs fail shorted due to electrical overstress.
Check your gate drive. It should be fast enough so you don't have excessive switching losses and high enough amplitude that you get good low RDSon. The threshold voltage on your FET can be as high as 4V, so you would want to drive it with a 10V driver. Not sure your Arduino is up to this task. Consider using a FET driver IC to provide better gate drive.
Check your drain voltage during operation. You have an RCD snubber, which is good, but be sure your voltage does not exceed the VDS rating of the FET.
Give 1 & 2, monitor the temperature of your FET during operation. It's probably getting way too hot.