You can set the temp for the unit to come on and it comes on and runs and cools for a few minutes then the compressor cuts off then the fan a little later. After a few minutes it comes back on and does the same thing. It never cools down to the set temp. The coils are wet but not frosting.

Possibly a tstat issue. Run system until satisfied, once satisfied indoor fan shuts down compressor still runs remove tstat face plate and see if compressor shuts down. Another issue could be a sticking contactor for compressor. With a meter you could check at the contactor to see if you lose your 24volts for cooling but contactor remains pulled in.

going out on a safety try changing compressor run capacitor.

Sounds like the thermostat, the way to verify is to unplug indoor unit, remove thermostat, take pic of wiring with phone or label the wires as to the terminals. Carefully twist the yellow,(Y) Red (R) and the green (G) together. Plug the furnace back in, if the unit operates normally, replace the thermostat. I am basing this on the fact that the blower speed is changing. This could also be an anomaly within the circuit board in the furnace section, not likely though as the outside unit (condenser) is not controlled n by the indoor board unless this is a heat pump. hope this helps. D

Make sure the filters are clean. Check what mode it is on as well. On heating, it is normal for there to be a delay before the unit starts up and then "run on" when turned off. Also check what temperature is set on it. If the temp is too high, the machine will cycle on and off as it thinks it has already reached the temp you want.

Assuming this is a heat pump I'm guessing you have a bad temp sensor. Your outdoor fan is likely controlled by a temp sensor so it only runs as needed during heating. Fan not coming on and compressor still needs cooled therefore compressor protection kicking in?????

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Could be but before that check if settings are working. Set the freezer settings to coolest and see if cools. If ok, set to desired temp.

your thermostat could be wired in to a energy recovery venilator which will bring the unit on for a certain amount of time every hour some thermostats have recirculate mode which will do the same thing to help filter the air

so your reported figures show that the unit is operating as planned
the closer to ambient temp the shorter it runs for
the further away from ambient temp the longer it runs for
when the desired temp is reached , the compressor will stop but the fan will continue to run to circulate air past the thermostat so that it will switch back on when the room temp reaches too high temp
and that is why it is recommended to set the temp at 25 C as that is the most economical temp to run it at

Check amperage while unit is running.

Without the model or make sound like a bad thermostat..check amps the the compressor and voltage... if the compressor is locking up or the thermostat cutting voltage to it..
Hope this helps