Normally, as the system heats up and expansion occurs, coolant goes from the radiator to the reservoir, raising its level some. As the system cools down, its pressure drops (it can produce a vacuum) and coolant goes back into the radiator, lowering the reservoir level.
OP points 2&3: With some system leaks, the reservoir level will drop significantly as its fluid replaces what has been lost. Sometimes, though, instead of sucking fluid from the reservoir, the system will suck in air through the leak. (Thus to check the level in the system, it’s necessary to look in the radiator–COLD ENGINE ONLY–normally the fluid will be up to where the bottom of the radiator cap goes.) It’s possible that your system is acting this way, sometimes drawing from the reservoir and sometimes not.
OP point 4: A cooling system pressure test will find most external leaks. In tricky cases, flourescent dye and ultraviolet light inspection can help.
OP point 1: To be certain the gauge reading is right, measure the temperature of the engine, preferably where the gauge’s sending unit is located. This may not be feasible without special tools (infrared thermometer or contact pyrometer probe). Assuming the gauge is right, I’d like to know precisely what a “few minutes” is. If it’s 2 or 3 minutes, I would suspect a blown head gasket to get to max on the gauge that quickly. If it’s 8 or 10 minutes, I’d suspect a flow problem–faulty thermostat or significant air pockets.
A few thoughts on the thermostat: all modern cars I know of have a thermostat rating of 195 degrees F. A somewhat imprecise but fairly effective way to check one is to drop it into boiling water–it should open right away. Do this for your own curiosity–as previously mentioned, it should be a pretty inexpensive part and I would not trust one that has been heat-stressed to work consistently.
Overall thoughts: if you’re lucky, the heart of the problem is leakage. Find and fix the leak(s), replace the thermostat, refill the system making sure there are no air pockets, and it may work fine. Of course, it’s possible there are further problems, but the above is at least a good start.