What is the point of the plug for the pilot jet.
There's a drilled passage connecting the area behind the main jet to the pilot jet area. With the plug, gas goes through both jets before it comes out the pilot jet. Incidentally, I seem to remember there's a foot in the bowl that holds the plug up, no worries if the plug feels loose in the hole. There's a small flat on the plug that ought to be turned the right way. The bowl also blocks the float pin, so it doesn't really have to be in tight either.
There's a BS34 version that doesn't use the plug and there is no passage connecting the two places. I had one on an SR250 (a fun little bike!).
I have a 1983 Heritage Special (USA) it has an idle sync screw between the carbs. ...How do I adjust it so I can adjust both carbs idle from the carb on the left side?
Just read the vacuum from both barbs and adjust the screw between the carbs till the right matches what you read on the left. It's pretty critical..you can feel just a few degrees difference. If it's off it will feel like it does when it's starting to run out of gas. Your vacuum gauge might just swing crazy, because the vacuum isn't steady and isn't supposed to be. Tighten a little clamp (vice grip, etc.) on the hose until the needle barely moves, then allow it 10 sec. or so to settle down before you take the reading.