The flywheel brake looks fine - so long as it disengages when you pull the dead man handlebar and then cuts the engine and brakes it when you release, that's it.
As far as I can see, the carb control springs look to be correct but you may be able to alter the running speed by moving the bracket that the bigger spring is fitted to.....giving it slightly more tension, perhaps.
Regarding the hunting, not every engine likes to run at a constant speed (although these ones are pretty good as a rule, to be fair) and if you notice the white air vane governor arm it may be moving a little as it tries to stabilise the engine speed - this is normal but the engine shouldn't chuff any smoke at all.
After all your work I guess the only thing you can really do is replace the carb but I doubt if it would be worth it, quite frankly, but keep the minimum of oil on the foam air cleaner element to prevent the engine running rich. Likewise, don't overfill the engine sump - a 10/40 semi synthetic oil is just fine, although a straight 30 grade oil is recommended. That's because a straight grade oil stays in viscosity better than a multigrade brew as air cooled engines run at a variety of temperatures.
Good work and well done!
John