Take a picture of the tip of the spark plug, I almost garrentee it's running rich... The bike would probably run better if you stop trying to make a problem out of small things. The carb looks fine, cap should be fine, and to get these things to start you have to slip the clutch since your bump starting the engine (to an extent), and the wire may need replacing but if it's still attached it should at least start (it's 7mm wire I use accel hitemp insulated wire seems to work better)
And get a spark plug NGK b6hs the last time I got some they were like 3 or 4$ it should make it run better