1) Is your main panel a 100 amp service or a 200 amp service? Do you have a sub-panel installed in the house or just the main electrical panel?
2) Is the AC located on the same circuit breaker as the microwave oven or located on separate breakers?
3) I assume that if you don't use the AC or the microwave, everything is normal and nothing trips?
4) Was there any recent electrical work performed in the house?
5) How old is the house? What is your wiring scheme? Metal conduit, Romex, Knob & Tube?
6) Are either the AC or the Microwave oven on a GFCI or AFCI protected circuit? Ground Fault or Arc Fault?
7) Same thing occurs if you use an extension cord and plug the window AC into a different circuit?
8) Does the branch circuit trip along with the main breaker or is it one or the other that trips?
Can you provide me with the circuit numbers of the microwave oven and the window AC located on the directory panel of the main electrical panel? The left side of the panel will be odd numbered and the right side will even numbered. Are the microwave and AC circuit breaker located on separate full size breakers or are they located on tandem (mini) breakers?
9) Do you have or can you borrow a 2-prong AC voltage tester or a voltmeter that is capable of measuring 120 and 240 volts AC? If so, are you comfortable to verify a few quick voltage measurements inside your main electrical panel?