Id love to see a combined heat and power system that burns natural gas to drive a heat pump compressor, and recovers the combustion heat. I wonder what multiplier it would get on a therm.
Gas furnaces are dead simple, and still break and require maintenance. Heat pumps, more so. Generators? By far the most.
Combining the three sounds like a huge headache for marginal efficiency gains.
As far as efficiency gains, we're talking 25-40% of the energy being produced as electricity and the majority of the rest being turned into hot water for residential use, either by one household or a whole city block. This doesn't sound like a terribly marginal gain. Specially if the energy production is done during the hours when traditional solar installations are not producing which are generally when folks are running their heating.
The working temperatures of ammonia are probably wrong for a heat pump, but perhaps something similar could be devised.