Shower.. Temp should be constant even with pressure fluctuations since they will increase both hot and cold proportionally.
Ice maker.. Cycle is as follows: Problems in (prens)
First: it runs on 120VAC.. Not battery power.
Temp of ice maker gets down to about 25F (Freezer not cold enough)
Thermostat closes and motor starts, also a heater
As the motor runs a cam closes a micro switch, bypassing the T-Stat
Motor runs till "Fingers" encounter ice, then stalls till heater melts it enough to push it out of the freezing tray.
Motor then continues to run pushing the ice out
As the cycle continues a 2nd Cam closes an adjustable Micro switch sending power
to the Water Valve Solenoid and opening it so that water flows (water path clogged with ice, bad solenoid, bad water line, water supply turned off, no water supply)
Ice is 'Scraped" off the fingers by the other fingers and falls into the tray (Often on the next cycle)
I would check for leaks outside. and check the cube size, Drop a Thermometer in teh freezer and make sure it is getting down to aroudn 20(F) or less.