Thank you for the update, FAST GRASS. While ring gaps for first and second rings are sometimes slightly different, I would not expect one to be double the other. Here are Honda's specifications for the GXV120:
Top and second ring gap: 0.008-0.016" new, service limit 0.039".
Oil ring gap: 0.006-0.014" new, service limit 0.039".
Hence I think the 0.010-0.020" Tecumseh specified would be the lower and upper fit-limits for new rings in a new bore. Do not go below the lower limit, or the engine might seize.
All in all, it sounds as if you should have a good solid engine there, at negligible repair cost, and you've had fun doing it. In my rather limited experience, Tecumseh engines do not usually lack mechanical reliability, though they may be rather prone to problems with the peripheral items, in particular the carburetor.
Please post the pictures when the photo-loading facility is back in action, FAST GRASS. It has been an interesting project and will be very useful in our archives when the record is completed by pictures - and perhaps some commentary on any tricky issues that came up in doing the work.
