You need to finish the replacement is all.
I'm in the process of putting in a new water heater so I have my hot water tank completely removed.
Now when I run the cold water in my sink, I am seeing some water come out of the hot water intake line. Additionally, now some cold water will out of the hot water taps if I open them.
Does this indicate some problem with the hot water getting into the cold line?
You are turning on both the hot AND cold faucets, which causes the water to backflow into the hot pipes. Connect the heater, but stop using the faucets until you are done.
Licensed residential and commercial plumber