Sounds like all is good without actually seeing the motor! So we are back to where you started concerning the prop, locknuts and backplate. Without purchasing those parts you can do what i've discussed in another thread. Take a piece of 220 grit sandpaper, fold and glue back to back using Super 77. Once this is dry, make a washer out of it and install between the backplate and the prop.

During this 10-15 sec run, did you advance the throttle? It may be too rich so close the hi end slightly. I've found that almost all of my YS's run around between 1 and 1 1/2 turns out.