After flushing toilet, hot water tap runs cold a few minutes before warming again... water heater new... notice the "out" (hot) copper pipe on water heater turns cool and the intake (cold) copper pipe warms after such

When you flush the toilet it causes a fluctuation in your water presssure for a moment. This alters the water coming into the heater, same for the water leaving the heater. Your heatyer has a dip tube that reaches down away from the incoming cold water to prevent this problem. Your dip tube may be broken inside the tank.

"someone suggested bad faucet (kitchen?) is causing "backflow" .

Looks like someone was correct. It's not "backflow" but a cross-connection in one of your single handled faucets that's giving you the grief. What you're going to hafta do is, one by one, shut the water off to each single handled faucet and give it a flush. Still running cold? Then that one's OK. Turn the water back on and move to the next one. Sooner or later you'll find one bthat the water doesn't run cold on. EUREKA! You've just located your culprit. Either repair it or replace it. Good luck, Tom