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This is a naive theoretical question but here goes. It seems to me
that many opportunities exist to use a Stirling engine to capture
energy where temperature differences exist already such as in a house
where the space is heated and cooled anyway for the comfort of the
occupants. So if you could build such an engine(s) in the wall of the
house to exploit temp differences between the inside and outside,
couldn't you then generate power at little or no extra cost (not
counting cost of the engine and installation) to pay for some (of
course not all) of the energy used to heat and cool the house?
Theoretically, how much energy could you extract? I guess the
question could also be phrased, how much energy (what percent? what
efficiency) does a Stirling engine consume to generate power? Would
your heating and cooling bill rise significantly to power the
Stirling engine or is the power generated coming at minimum cost?
Thanks.

There have been Stirling engines built that operate on temperature differences of a few degrees, BUT the power output is barely able to overcome the friction and keep them running.

CHP, Combined Heat & Power essentially acomplishes what you suggest in a very different way. The gas normally used to heat a house is used to run a Stirling engine and the waste heat is used to heat the house. The mechanical power output is used to run an electric generator to light the house as needed and pump the rest into the local power grid!

Your idea assumes running the Stirling engine with the house heating the hot side and the outside air provide the cooling. Basically the temmperature difference is not adequate. If it would work it would be in the class of perpetual motion machines. You can't get any more energy out of a system than you put into it to start with, hence No Free Lunch!