What are the two tubes connected to my refrigerator?
Sounds like the tubing is reversed, there is a valve that will hold pressure from one direction only. I have never seen gray line connected to main supply. 
There really is no Hot water connection to a refrigerator. Although hot water will freeze faster than ambient, that would defeat the purpose of a refrigerator, by pumping in hot water. Besides the water doesn't stay on long enough for it to work well. The gray line most likely goes in a tray under your fridge. The drain from the in door water tray has to go somewhere. So I have seen them in a tray so that the heat from the compressor and fans will evaporate the wasted water and melting ice cubes.