For the time being you could just go to a mower shop and buy a new diaphragm cover and new digaphram and try it.
If the cover has a crack/split/hole the govenor will not work propperly.
Got a pic of the diaphragm itself?? I might be able to give some advice. Quite often they are not sitting right when the cover is put on and they tuck up inside the body of the carb. The motor wont run right as its then a vacuum leak. Then the diaphragm sets like that. With some warmpth and gentle stretch some of them can be saved. The type, thickness and profile of the diaphragm and the tension of the springs etc is what makes it go right. Starting from scratch with a box of odd carb bits can be a dead set nighmare.
The fuel gauge caps. They go white if theres any ethanol in the fuel. I pulled one out of a mower i got here and the plastic was all cloudy and soft, that soft that a tear drop had formed on the end of it. Had ethanol fuel in the tank, stinking stuff. A day in the open air and it was hard as a rock again but still all cloudy.
Make your sparks work first. Check things, get new plug boots. The screws in the HT leads might be a bit corroded, you can clean them up with some sand paper. 2 new spark plugs. See if you can get some good saprk again.
Gont be distressed about the problems. You have the biggest problem sorted, that is the catcher, looks like that one is in very good order. Ive seen a few twins up for sale, 9 out of 10 dont have catchers.