I'm happy to be corrected, but my recollection was that they provided some specifications, what we would call SWaP today, and some sketches to support, and Taylor designed, built, and tested the engine.
The funny thing is that today the separation of aircraft and engine design persists.
This in a new term to me. I believe it means Size, Weight, and Power. Specifically as an optimization problem. The modern version includes cost as well which was probably less of a concern with the Wright Flyer.
No, they designed the engine and hired a machinist (Charles Taylor) to make it.