I have 3 bathrooms. When I flush any one of the toilets, the cold water pipe at the back of my Washer vibrates just during the flush and beginning of refill. I can make the noise go away by turning on the cold water tap at the bathroom sink. Noise stops while toilet finishes filling. I've also noticed that when a toilet in the house is flushed, water pressure at the kitchen sink is almost zero. Started 3 weeks ago out of the blue.

You have either a loose washer somewhere or a faulty pressure regulating valve. Since you didn't mention a noise associated with this vibration I would think a faulty PRV.

Thanks, the noise accompanying the vibration is a low drone sound, deep humming, 4-5 seconds