I can't tell which one the water is flowing through)
If hot water is flowing from the tank, then cold water will be flowing into the tank. The only time cold water could enter the tank with no hot water flowing to home fixtures would be if there was a leak from the tank, or if the T&P valve is allowing water to exit, both of which I would anticipate would be noticed by you.
The T&P valve is a safety device which will normally only open if the internal temperature or pressure of the water is higher than the valve's release settings (usually 150 psi and 210 F). The T&P valve is often located at the side of the tank near the top. It will typically have a plastic pipe threaded into it which will direct any discharge down to the floor, or the pipe maybe extended to a floor drain if one is located nearby.
the blue copper pipe feels hot.
When no hot water has been used for some time, the cold inlet pipe will usually slowly become warm as the hot water within the WH will transfer heat through conduction and convection.
Also, when cold water is heated, it will expand. As water cannot be compressed, the expansion will need to flow somewhere so with no hot water being consumed, the expansion will flow backward through the cold water inlet to be absorbed by either the municipal distribution system if on city water with no backflow prevention, or into an expansion tank if there is backflow prevention. If your water source is your own well, then the expansion will be typically absorbed by the pressure tank.
I can definitely hear water flowing in or out of the tank
Is your home connected directly to any other homes?
I reside in an end unit townhouse so I am directly connected to 1 neighbour. I can occasionally hear my neighbour's water flowing, which I sometimes think is water running in my home.
I can also sometimes hear water within my gas WH making a slight hissing sound while the burner is operating and water is being heated.