Well as you say you have found a small air leak in the carb rubbers, this would draw in the brake cleaner as some of it will quickly turn to gas even if you are just spraying around the carbs but also the rubbers to the airbox. Either of these would lead to a rise in RPM. Take it step by step and after fitting new rubbers, try the brake cleaner test again. If it still rises i would suspect its entering via a leak on the airbox rubbers side. If your idle is stable (when warm) it is unlikely to be leaking serious amounts (or any ) air in where it shouldnt. The backfiring on throttle shutting from raised RPM will be air being drawn in via the air leaks you have found after the throttle butterfly, in the rubbers, so fitting new ones should eliminate that. As for play in the throttle plate shaft, some wear is likely to be there, but as long as when you blip the engine it always settles to a steady idle within a few seconds, not maintaining an elevated idle or erratic idle then there is nothing to worry about.

Sorry cant see your video as i dont have Veed. Post it as something standard like MPEG4, AVI etc. If you still have the carbs off it would be a good idea to go through them and replace whatever gaskets and seals you can. Probably a good idea to replace the float needle valve as yours are bound to be old and who knows how long they will be available. Either way check the float height (refer to manual) as many people overlook this and then have fuel mixture issues on all sorts of petrol engines! Also make sure the carb airflows are balanced and idle mixture screws adjusted and it should tick over like a Swiss watch (like mine

). I extended my idle screws to make them easier to adjust by hand in the middle of the hot V twin, as shown in a previous post.
Happy twiddling !