Plumbing – Why do the water pipes jump when running water

The problem is your expansion tank, or lack thereof. Your home has a check valve somewhere in the system, possibly included with the pressure regulator or as part of the municipal water meter. When your water heater turns on, the heating of the water causes it to expand, and that pressure has nowhere to release until you open a faucet. What you need is an expansion tank installed in your water lines. If you already have one, then it has failed (they usually have a bladder separating the air from the water, and this can fail over time or if not properly maintained). Installing a tank isn't too difficult, but will require shutting off the water to your house, cutting your pipes, and adding a T to your water line. If you have an existing tank, then they are often threaded and are simple to replace (the connection is often threaded and can be screwed off). Note that if you don't fix this, valves in your toilet and other fixtures will start to fail, or the TPR valve on your hot water heater will begin to leak water.

Likely problems

1. Over-pressure from the street. Solution is installing a pressure reducer just after the main shutoff.
2. Missing or leaking "water hammer arrestor", which is a vertical pipe with air in it or a tank with an air bladder, meant to soften sharp pressure changes.

Water temperature is not a likely candidate except at the extremes. If you have a local tool library, or perhaps home store, they may loan out pressure gauges.