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That is indeed a complex situation. We should check a few basics first. Is the air filter clean, and do you have furry pets in the house? A dirty heat pump coil that fits one either end of the furnace will block air flow and cause the heat pump to go off on its high pressure cut out switch.
It is also probable that after 5 years the radiator (outside unit coil) is clogged with dirt deep between the fins where it cannot be seen, after 5 years it will need cleaning if it has not been cleaned in the last 5 years.
None of these however address the back up heat not coming on, we can address that separately.
You can buy a condenser coil cleaning kit at most larger hardware stores or departments... be careful not to let the pets eat the resulting foaming cleaner.
Let me know about all that, we can go from there.
Thanks, ***** ***** when the finned coil on the outside was cleaned please. That will be a problem if not cleaned, it is probably not the entire problem, but could be a major part of it.