Lynn,
The shower issue sounds a lot like air in the water heater tank. If it is, this is easy to fix. The air in the tank makes it behave partly as an accumulator, so try this before you add one....
With the heater off and the pump on, open the taps for hot as fast as you can. This will work as well on city water as on your pump. If you get a burp, you have probably fixed it.
With the air in there and on the pump, there is a lot of pressure fluctuation. As the pressure is going down, the tank puts more hot in the system. When the pump starts, then the cold gets the lead.
As for the ice maker, it sounds like the fill valve is hanging up, but without make and model I can't get any closer.
Matt