Hot water line. If you did hook it up to your cold water line make sure your dishwasher is set to heat the water (some models have a selectable setting for this.)
It should be hooked to a cold line.
Install on cold water line.
I think the only reasonable answer can be that the cold and hot water inlet/outlets are reversed. your hot water line is attached, at the heater to the cold water inlet or your hot water outlet is too short inside of the tank
YES
The hot water line should be connected to the left and the cold water line should be connected to the right for proper functionality.
Odds are that the cold water lines to the faucet you are trying to use are shut off but the hot water lines coming from the water heater are not. That is why you have hot water but no cold water.
Hot? Cold? return? fire suppression ? Non Potable?
No, cold water is not hot. Cold water is at a lower temperature than hot water.
I am not a plumber, but am in the process of installing a water line myself...and all of the research I have done says the water line should be hooked up to the cold water.
Not being able to see the pipe and not knowing where it is, it's kind of hard to know. If it is a cold water line and you have not shut off the hot water tank, the pressure from the hot water will back flow and hot water will come out of the cold side.
Check piping. Make sure cold line going into cold side of water heater and hot side coming out hot side. Cold feed for water heater should come off main cold line. Make sure valves are open to and from water heater.