Why do you lose hot water in the shower when a tap is turned on elsewhere in the house?

this is probably due to improper sizing in your water lines. The lines don't have enough volume to handle everything that's running. Water tries to take the easiest path that it can so if you turn on something that is closer to the water heater or has less bends in the line the hot water will go there. You can increase the line to the tee to a minamum of 3/4".