For anything with sophisticated electronics inside, particularly electronics that is taking the place of a mechanical on/off switch, it is important that the computer inside is started up cleanly. And in this context the microwave and the TV also have microprocessors inside just as computers do.
Every computer will have circuitry inside that lets it turn on, lets the voltage rise to a minimum level, and then sends a signal to reset the computer to a known condition and let it boot up.
Sometimes this startup circuit is not very conservatively designed, with the result that short power fluctuations (glitches) can lower the voltage enough that the computer malfunctions, but does not lower it enough for the power-up reset circuit to be properly activated. That can cause things of the sort you see.
There may be an intermittent or high resistance connection somewhere in the electrical feed to the trailer. If in addition to the behavior you saw, there is also light dimming or flickering, changes in the sound of refrigerators or fans, etc, then this is very likely the problem.
This should not have any direct connection with bad cell phone service, so that is probably just a coincidence.
The other possibility that would also explain the cell phone problem would be emissions from an alien spacecraft hidden under that space in the park.