Regarding the a/c button, if you have the selector on defrost, the a/c should stay on. If you turn the a/c off, and turn the selector to one of the other settings (such as floor or vents), does the a/c turn off then?
Regarding the heater temperature, if the fan is blowing normally, it is likely either the thermostat is stuck open or the heater core is clogged, either internally (not allowing the coolant to flow through it properly) or externally (not allowing the moving air to pull the heat off of it properly). Does the car warm up to its normal operating temperature? If you have a temperature gauge, check to verify that it warms up to the middle of the normal range. If it does not, chance are good your thermostat is stuck open. If it does, then the thermostat is probably working correctly, and it is more likely that the problem is with the heater core.