Nest in dual fuel system frequently stops generating heat

[Mike_Schaeffer]

We have a Nest E thermostat, along with Rheem heat pump and auxiliary heat is a natural gas forced air furnace that originally was the only heating system.  The system typically works fine, but it irregularly stops generating heat.  The fan will keep operating for hours, but the temperature does not rise, and the air coming out is definitely cool.

Works fine tells me that the wiring and programming isn't dead wrong.  The heat pump was added to the system by a professional HVAC outfit, not DIY.  They have been back twice and just seem to doubt our experience, lecturing us that heat pump hot air is not very hot, and it goes into a frost cycle, all that.

Wiring - we have wires connected to Y1 heat, G fan, R power, W1 alt heat, C 24v common, *ob heat pump.

Any suggestions for programming missteps would be appreciated.  I also have to say I don't think I have ever seen the word "aux" on the display signifying the aux system is operating. I'll say that the safety temperature is set to 40 degrees F, but is there a different parameter for the temperature at which to use Aux heat?  The text associated with safety heat sounded like it was not the same thing.

[CoolingWizard]

@Mike_Schaeffer 

in a typical heat pump arrangement the air handler normally has a set of electric heat strips that are actually the auxiliary heat. In addition to auxiliary heat, they can be wired as emergency heat.

now in your case, you have a gas furnace and this is recognized as a dual fuel system and unless you specifically wire W2 to your gas furnace, you will not have any auxiliary heat for your heat pump.  

when your heat pump ice is up outside, it must switch from a heater to an air conditioner to get the warm air in the house outside to melt the ice. Now, during this time, the outdoor unit will signal the indoor unit to go into auxiliary heating mode.Now in your case, we have to make auxiliary heating connect to your gas furnace. This is not a technically correct HVAC configuration but it does work. You eill need a W2 wire in your thermostat that connects to W1 or W in n your air handler/Furnace.  The defrost signal from the outdoor unit will need to connect to W1 terminal for n the furnace as well.  
with this configuration, your gas furnace will be turned on anytime. The outdoor unit goes into defrost mode, or anytime the indoor thermostat detects. The heat pump is not keeping up with the demand for heat. if I were your HVAC contractor, I would recommend putting in 15 KW of heat strips in your air handler and let that be your auxiliary heat. Then allow your system to be a true dual fuel system and at some point around 23 to 25°F, you switch from heat pump to gas heat. 

AC Cooling Wizard.