The cyclic speed variation is almost certainly hunting, and can be cured completely just by turning the mixture screw anticlockwise slightly (one eighth to one quarter turn probably, but just adjust it until it doesn't hunt and runs nicely). If the puff of smoke is black, it is just petrol smoke caused by the governor suddenly slamming the throttle all the way open (acceleration enrichment), and if so, there is nothing wrong with your engine (which is quite a nice one). If you post a picture of the carburetor we can probably point out the mixture screw to you. The spring-loaded screw in your last picture, above, is the idle speed adjustment and is on the carburetor, but not the same side as the mixture screw. The lever the idle speed screw is on is part of the throttle butterfly, and it is moved by the governor. The governor moves the throttle to control engine speed. When you adjust the mixture, watch the lever the idle screw is on: as you reduce the leanness of the mixture, the lever will move less. When the mixture is close to correct, it will move only slightly and seldom unless you put load on the engine or move the speed control.
Based on what you have posted so far there is no reason to replace anything in your engine, and it will probably be more reliable if you do not do so. However if the puffs of exhaust smoke are blue, rather than black, your piston rings are probably on the way out. I suggest you adjust the mixture which should get it running properly, then assess whether it smokes.
