heatpump water heater caused house no cold water?

Upgraded gas heater to Rheem 50gal heat pump water heater but NOT POWERED ON yet. Turned on cold water inlet heard the water goes in. Turn on bath hot faucet, air came out then dirty water then clean water. Turned on kitchen hot faucet, lots of air with splashed water so I turned it off - along
with valve at the tank inlet. This time faucets cold water work as usual.
Next morning I turned on bath/kitchen faucets again, both has low stream water flow either on cold/hot valves! Tank inlet ball valve is on, Backyard faucet is ok cold water pressure is ok. Turned a bit T&P valve water does come out - means tank has water. I do have a vacuum relief valve(Homedepot 300702609) at the tank cold water line, no expansion tank.
How can a water heater cause rooms faucet give very low stream of cold water? It should do nothing with it. And while cold water inlet valve is on, hot faucet shall come out air then water continuously. House is old 60+ years.

 

Your aerater on each faucet is stopped up with rust or lime from disrupting the piping. With aerater off flush hot & cold water lines.