The answers to your questions are fairly complex. The answers given here are thus VERY simplistic. Full answers can be found in the ASTM standards that have been accepted by the FAA.
so, with that in mind, here are some basic answers to your questions. Since you mentioned "airplane" these answers will be from the viewpoint of certification of a fixed-wing airplane. Answers may be different for other classes (powered parachute, weight-shift, glider, etc.)
1) For SLSA certification the airplane engine must either meet ASTM standard F2339 or be type and production certificated by either FAA or JAR. I don't know of any Honda engine that would fit this criteria, but there may be a Volkswagen derivative that does (Limbach??).
2) There is no specific requirement in the airplane standard for the propeller to be certificated. There are performance limits for the propeller called out in the standard.
3) There is a great deal of testing called out in the performance standards, including loading, performance and stability. Far too much info to cover in this forum. In order to get a full understanding of the testing required one would have to look at ASTM standard F2245.
The potential manufacturer will also have to meet standard relating to quality assurance, production acceptance testing, maintenance and inspection procedures, aircraft operating instructions, and several others. The list of applicable standards can be found here.
If you're really serious about wanting to pursue this, you really need to spend $75 and join ASTM. This gives you full access to all the applicable standards, and is much cheaper than buying the standards individually. The standards are not public domain documents and they do not belong to the FAA or government, so you cannot access them without joining ASTM or purchasing them individually.
Hope this helps!