Unit shuts of at 70 degrees F. Setpoint of the thermostat is set at 56 degrees F. I get the impression that the temp sensor might be out of calibration. The unit model is a ChillR plus and sreial number is 07200-088

Some thermostat will run a little longer than set point to maintain a space temp but depending on the age of thermostat probably need to replace it.

Some thermostats have a swing setting that can be adjusted. A swing is the setting when you want the furnace to stop or start from the desired set point. For example, you set your thermostat at 70 and the swing is set at 10 degrees. Then the furnace will turn on 5 degrees below 70 and shut off 5 degrees above 70. Check you thermostats manual if it has this option.

Have you cleaned the air return filters ? Wash them, don't vacuum or brush them. It damages the filter screen. Sometimes a thermostat will be out by a few degrees. If the filters are clean, try turning the temp down a bit to 65 degrees and see if the air con shuts down. ( If your room temp is showing 70, it should switch off the machine) If it doesn't turn off, keep reducing the temp by 5 degrees at a time and wait a few minutes each movement to see if and when it shuts down. If the machine keeps running, I would suggest your thermostat is shot and you will need a new one.

If you are talking about the difference between heating mode and cooling mode and the room setpoint, you can not change it. It comes from the factory with a 3 degree difference. In other words if you raise the heat setpoint until it is within 3 degrees of the cooling setpoint, for every degree you raise the heat setpoint the cooling will raise a degree.
About all you can do is increase the distance between the heat and cooling setpoint. To do this either lower the cooling setpoint to 3 degrees higher than the heat setpoint you want and then increase the cooling setpoint the desired setting. For example heat and cool setpoints are 67 and 70 respectively, and you want 65 and 73. Lower the cooling setpoint to 68 and then raise the cooling setpoint to 73. Now your heating will come on 3 degress lower than 68, the 65 you wanted and cooling will come on at 73. Remember, the thermostat new setpoints will turn on the cooling at 75 and turn back off at 73 and heat will turn on at 63 and turn off at 65.
You can perform the same using the heating setpoint, just have to look at it backwards. If you need to lower the difference I recommend a 8000 series thermostat. They are the most functional thermostat that Honeywell makes, be in mind they are about 3 times more expensive and many of the setting require a well trained tech to interpret and set properly.
Hope this helps.

There's a "throttling range." The unit starts (say, in heating) one degree below setpoint and shuts off two degrees above set point.

You can change the "temp" but you can't change the throttling range.

If your thermostat has an emergency heat setting this means you have a heat pump system.
Yes you need to set the thermostat to heat for heating the house.
The emergency heat setting should never be used unless your outside unit is not functioning.
The emergency heat setting overides the heat pump (the outside unit will not be energized).
The emergency heat setting will run the auxillary heat (heat strips) only.
When the thermostat is set in the heat position the heat pump will run to heat your house.
If the heat pump can't keep up (extreme cold) then then the auxillary heat will kick in to augment the heat pump until it satisfies the thermostat settings. If your room temperature gets 2 degrees below your thermostat setpoint, your auxillary heat will kick on. Example, say you are going to be gone a couple of days and you turn your heat down to 60 degrees. When you return and it is 60 degrees in your house, you turn the thermostat up to 70 degrees. Since you are turning the temperature up 2 degrees or more (10 degrees in this case) from the 60 degree room temperature then the heat pump will come on and the auxillary heat strips. The unit will continue heating like this until the temperature in the house gets to 69 degrees and the heat strips will turn off and the heat pump will continue to run until the 70 degree setpoint is reached.
Hope this helps.
SeagullAC

Most thermostat have a dead band setting that you can set yourself.
Maybe your dead band is set 1 or 2 degrees above setpoint so unit would run to satisfy deado band

If your heat anticipator in the thermostat is bad or set at the wrong amperage, it could do just as you describe. If the air filter is dirty or the ducts are restricted, the furnace could shut down on high limit. I would check the air filter and replace if necessary. Make sure all registers are fully open and last, change the thermostat.

check to differential setting in the thermostat and are you saying the temp in the room and the set temp is being read from the thermostat or is there another thermometer in the room your looking at

sometimes the furnace does a better job than anticipated, i wouldn't worry about it, especially if the furnace shuts off.
the setpoint is the temperature you want it to be
cycle lenghts are determined by how much recovery you are requesting. for instance, in the morning when you wake up, you want the temperature to be at its setpoint. what the thermostat will do is make the furnace run a limited lengths during the recovery process.