While feeling around down there, the cold should have no water running in it, because the stop valve is off. I guess as another test for an unlikely condition is to turn off both stop valves, and make sure water does not come out of the faucet.
One possibility is that the hot from the wall is supplied through a thermostatic mixing valve. If that valve was adjusted too cool or had failed, that could explain the symptoms.
If you don't have a basement, I don't know where I would look for such a valve. I would presume close to the lavatory. Maybe behind an access cover? I am not a plumber.
I have no insight on what that loop is. Your topside photo made one theory go away.