AC not turning on until well over 4 degrees of set temp
Thermostat controls will always have some hysteresis between the ON detect temperature and the shut back off temperature. This is necessary so that the AC unit wouldn't be toggling on and off all the time at a rapid rate. 

More exact control is possible with a more expensive temperature control but that normally requires that the AC unit have an ability to operate an a variable mode other than straight OFF and ON. 

It is possible that the thermostat that you have is really cheap or out of adjustment and leads in part to the behavior that you are experiencing. The wide range behavior can also be caused by a situation where the thermostat and the temperature setting controls are mounted separately from each other or not in optimum placement. There is also the possibility that the scale of the thermostat setting and the temperature monitor are not in agreement.