If the flap is down, it is unlikely that the water is coming from the tank. Does the level in the tank drop when the bowl fills up? Is it possible that the drain (waste pipe) is backing up? Does the water rise in the toilet if you run water elsewhere in the home? Could the waste line be frozen?
At temperatures that low, many possibilities arise. There should be a valve to shut off water to the toilet directly under the left side of the toilet. I am sorry for all the questions but it is difficult to understand the problem. When you say the 'water keeps filling the toilet" are you talking about the tank or the toilet bowl (porcelain part one sits on). I am not a plumber but I have experience with a lot of these items. If it is the tank that is over filling, the valve mechanism located in the tank must be broken. If it has a floating ball, is the ball floating? Also, there may be adjustments on the valve on the left to lower the water level. Write back if you can answer some of these questions.
At temperatures that low, many possibilities arise. There should be a valve to shut off water to the toilet directly under the left side of the toilet. I am sorry for all the questions but it is difficult to understand the problem. When you say the 'water keeps filling the toilet" are you talking about the tank or the toilet bowl (porcelain part one sits on). I am not a plumber but I have experience with a lot of these items. If it is the tank that is over filling, the valve mechanism located in the tank must be broken. If it has a floating ball, is the ball floating? Also, there may be adjustments on the valve on the left to lower the water level. Write back if you can answer some of these questions.