Think you need to start at the beginning. You need a $10 adjustable spark tester and a compression tester. The compression tester needs to be the screw-in type -- might be able to rent or borrow one if you don't want to buy one. You need to measure spark (try for a half inch and work back until you find spark -- 3/8" or more is what you want). You want compression of 120 or 125, give or take -- needs to be close between cylinders. The carb needs to be fully disassembled, soaked and cleaned with all passages cleaned out. The fuel pump diaphragm needs to be complete (no holes or tears) and supple (no set). The motor must warm up but not overheat -- around 130F or so surface temp on the top of the head, no higher than 160F. An infrared gun is a useful tool. If the impeller is more than a couple of years old, it probably needs to be replaced.
Fuel needs to be fresh with no sediment or water. The tank vent needs to work in order for the fuel feed to work, and the tank pickup has to be clear. Fuel mix is 50:1 using a good two-cycle marine grade oil.
Test the motor using the throttle control after doing the link and sync procedure described at the top of the forum.
On you pic -- the plate on the front of the carb throat is the choke plate, and is open in the pic (and appears to be discolored). After soaking and cleaning a carb, the brass will be bright and shiny -- same with the idle mix needle. You are operating the carb throttle plate linkage with your finger. The linkage should begin to move when the roller contacts the mag plate under the flywheel as you throttle up (also referred to as the timing plate or throttle plate). You can see the pick up point in the rubbings on the brass pick up -- should be an indentation or other mark in the vicinity. You should be throttling up using the twist grip or remote control lever (observe movement of the mag plate and carb linkage while throttling up). Additionally, it appears in the pic that the compression plug (domed plug) on top of the carb might be missing, although might be just the angle of the shot. If it's missing however, you will not be able to get the motor to idle and run. The area under that plug is part of the idle passage. It's also possible that the tip of a needle is lodged in the opening under there, either from the current needle or from an earlier one. If so, you won't be able to get the motor to idle.
Throttle position in the pic appears to be slightly advanced, maybe fast idle setting.
(sorry -- long response.) You may well have covered some of this....
