The answer is not a cut and dry one in your case. The troubleshooting is a process. You will need an understanding of a multiple light parallel circuit. You will also need a proximity type voltage tester and possibly a VOM.
Assuming the voltage feed starts at the switch (not always the case, but normal) you will need to verify input voltage at the line side of the switch with the switch in the off position. If you have voltage there, turn the switch on and verify voltage on the load side of the switch. At this point, a VOM is handy to test voltage across the hot and neutral and/or ground. If this looks good, proceed to the closest light fixture, and with the switch on, test the center hot tab in each fixture with the no-touch tester. In your case I would think you may not see any voltage based on your question. This could mean that you have an open neutral, an open hot, or at worse, a shorted hot to neutral/ground.
Since there are several possible reasons for your condition, it would take a lot of tutorial to explain every possible scenario. Assuming you do not see voltage at the fixtures, the basic technique for troubleshooting will be to start at the last known verified voltage point, then follow the wiring and check all connections in the junction boxes. Visually check the bulb sockets for broken or shorted metal tabs. This should be done with the power off at first, looking for obvious loose or disconnected wires, then with the power on using your voltage tester. Since this condition occurred after you adjusted the height of the sockets in the fixtures, I suspect tension on some wire may have pulled a wire out of a socket base or out of a wirenut in the j-box. Unfortunately, most can/pot lights have a built in j-box attached to the top of the fixture. This necessitates dropping the fixture down below the ceiling to access the j-box, or getting access from above. (attic).
Basically, you are following the circuit looking for an open or short, just like following a hose, looking for a water leak or stoppage.
This is not a hard job, but extreme caution must be taken when testing energized circuits. If you do not have good electrical skills, the proper test equipment, or a logical understanding of switched paralleled circuits, then this job is better left to a pro.
Maybe one of my buddies here on SE can add a good graphic showing this type of circuit and the test points. A simple line drawing showing the switch and junction points would be a great edit.
You say that after the rain and some period of drying the GFCI returned to normal working condition. Indicates your problem is "outside" related or influenced. Take a real good look around the outside of your building. Look for any plugs that may be catching moisture from above as the rain comes angling in.
Often as an electrician I come across previous wiring in homes where they jumped the inside GFCI to outside plugs to save money on spec homes. Home owners complain that outside plugs are not working yet breakers are un-tripped. I always say go look for a GFCI in one of the bathrooms, if it is tripped then reset it.
So if you have a non-contact tester you could turn off the breaker that supplies the inside GFCI, and test to see if any plugs on the outside of house are off. If so, make sure that any waterproof covers are in good state and replace them if they appear to be in poor state. If you have a cord plugged in for any length of time you should have a "in-use cover" installed.
Even a small amount of moisture leaking over the receptacle will cause small current leaks to neutral and therefore trip the GFCI and you will not be able to reset it till it is dry. Also any foreign materials will exacerbate the situation. So surfaces should be dry and clean on the receptacle.
Best Answer
An RCD (in any form) is always looking at the current going out vs the current coming back, which SHOULD be zero difference, so if they are different by the amount of your RCD (usually 10 to 30mA in your part of the world), the device trips. It might be that your type of meter is one that measures by looking at the potential difference by putting on a small load, so the act of trying to read voltages was a condition that the RCD interpreted as a short.