I haveNow that you've confirmed it's nothing to completely agreedo with Daniel Griscom. And, please slow down & stop over-thinking the situation. Breathe. You never had an expansion tank before, so why would you need one now. Everything worked beautifully before, so why not restore what you have back to proper operation.
I've never had a Pressure Reducing Valve, but the Water Heater should never overpower its supply line, reducedT&P valve or not. And, as proven by your hot water faucet opening, you've got a ticking time-bomb T&P valve.
Yes, Water Heaters have literally exploded, shot through walls & into neighbor's homesHeater & shot straight up through roofs to still go another football field higher. They've killed instantly in all of those real life examples.
Replace the T&P valve ASAP as your next & first step, whether it needs it or not. If you're goingonly to hesitate, then at least open the T&P valve next time, instead ofdo with the faucetPressure Regulator.
After the cheap & easy replacement of the T&P valve. Then, you can experiment in safety, IF the problem still exists. Other than that, I think the drain & refill idea is totally unnecessary. When the rumble starts, just turn off the cold water supply valve at the Water Heater.
If that stops it, then I presume your correct course of action is to adjustdefinitely play with the Pressure Reducing Valve up to a slightly higher pressure until the rumble goes awayRegulator's adjustment when it happens again. But, if the valvePressure Regulator is acting up or not working normally, then I'd order a rebuild kit & rebuild it.