Hello: If you put a working check valve on the cold inlet to a heater, what happens when water is heated? It expands as it warms and normally that expansion simply backs into the cold supply. If that can’t happen, pressure builds up to the relief valve setting, (150 psi) and works the valve. Doing this stresses all the hot piping and heater and will lead to failure sooner than it would have failed otherwise. You can add an expansion tank, but it will fail in time too and usually simpler systems are better.
If there is a cross connection between hot and cold, it actually could prevent damage to the hot side, if a check valve is installed in the cold supply to the heater. So, fix the cross connection and damage the plumbing 😕 If incoming pressure is over 80 psi, you do need a reducer and an expansion tank somewhere in the cold piping.
Is that answering your question?