We have been getting loud vibrations in the pipes (sympathetic vibrations) that last anywhere from 5 seconds to 10 minutes. The severity of the vibrations varies as well, and all are most noticeable at the lines connecting the refrigerator, the toilet, and the washing machine. I replaced everything inside one of the toilet tanks (flapper, gaskets, pump, etc), but the noise has continued. I was going to replace everything in the other toilet tank, but I heard our hot water tank making the popcorn sounds usually associated with sediment. Last night after I used quite a bit of hot water, there were several vibrations with long durations. The house and hot water tank are ten years old.
Is there a relation to these vibrations and the hot water tank? What is the best next step: replace everything in the other toilet tank; flush the hot water tank; something I'm not aware of; screw it and call the plumber?

Gas or electric water heater, My bet's on gas so I'll run with that.
If you do not flush,(not drain) your heater on a regular basis it will build up mineral on the bottom of the tank. When the flame comes on it turns the water trapped under the mineral to steam and the steam escaping is what you hear. Let me show you how to flush your heater. Attach a hose to the boiler drain at the bottom of the tank. With the pressure on, open the boiler drain and let it run until the water runs clear. You will see a spurt of red,(rust) and some white grains,(lime or calcium carbonate). The whole thing shouldn't take more then a few minutes. Don't forget to flush it out every month. Your heater will thank you for it. You might want to consider cleaning the screens on your aerators if you have a high mineral buildup. If the problem still persists after you have flushed, you will have to drain the heater, remove the boiler drain and go up in the bottom of the tank with a wood or plastic rod, (not metal) and try to break up the mineral and flush it out. Good luck and thank you for rating my answer, Tom