How do Stirling engines work? I understand the heating and cooling of air, but how much faster (or more force, I'm not sure which to use) does the piston move per degree Celsius that the temperatures differ?
The amount of force generated by a Stirling engine for a particular temperature gradient is different for each engine and depends on things like the particular engine configuration, materials used, volume of the pistons, stroke length, ... As for the speed or frequency since its cyclic, this will also vary by just as many parameters.
In general the larger the heat difference available the 'better' an engine will run, by how much is something for the individual engine product manuals.
For a given temperature gradient (say between an underground volcanic heat source and the open air) a trade off needs to be made between speed and force. An engine could be designed to be more powerful by making the pistons and cylinders really large to maximize the expanded gases pressing on the piston, or really fast by making pistons and cylinders much smaller so that the smaller amount of gas inside the chamber finishes expanding much faster for a new cycle to begin.