Kitchen Large Electric Appliances - Refrigerator/Ice Maker

I just moved into this rental about three weeks ago and last week I noticed my refrigerator was not making ice and the lever on the ice maker would not go all the way down into the on position, so I pushed it up into the off position and left it there. A few days later water started flowing out of the ice cube dispenser and we couldn't shut it off. I pulled the refrigerator from the wall to turn off the valve and found that the line was not connected to the valve but to a line coming out of the wall that I believe is connected to the reverse osmosis system under the sink. Unfortunately the old owner has left no manuals to anything and has no idea how the plumbing was set up. So the only way to get the water to stop flowing was to unplug the regrigerator and everytime we plug it back in the water starts flowing. Can tell me what the problem might be and how to get it to stop. I don't know if it is mainly a plumbing problem or a problem with the refrigerator only as I am also having problems with what seems to be a blockage in the plumbing.

If the water starts flowing when you plug it in, then the problem is not in the plumbing, but rather in the ice maker itself. Something inside is telling it to fill, but it shouldn't. Sometimes total replacement of the icemaker is easier than trying to get one repaired. You can check into it via the manufacturer's website, but quite often they are held in with two screws on the bottom and clip in the top.