Get the kit and install the electric choke. As far as the "quadrajunk" you actually have the correct carb for your motor. I can't tell you how many people want better performance, so they slab a bigger carb on there. Then they complain that they can't get it to idle right or other issues. I...
Not possible to leak fuel from that high up on any outboard. The only way it could, would be if the top seal was bad. Even then the crankcase is mostly under a vacuum, so it doesn't come out of that area.
Had two with similar issues at the shop in the last couple years. Both had everything replaced, caps rotors wires, fuel systems checked, before they got to me. Turned out the sparkplugs even though they looked almost perfect. Replaced them myself and did sea trials on both, Perfect! Sometimes...
I'd say that's just condensation from starting. When the air runs past the plates it's normal. When the motor warms up it don't do that much. I had a car that would freeze up when colder out and the throttle would stick due to the condensation, making ice. After warm up it was fine.
In early winter, we clean all the snow from the car before heading out. By this time of winter, we start the car, warm up a bit and say f**k it and go.
You could try tuning the key on, no cranking. Turn key off and back on a few times to build fuel pressure. Then start. A weak fuel pump/system may not build the pressure needed. I've done this on fuel injected cars to get them going to get them to a shop or home. Sometimes it work's sometimes it...
If the flywheel is just going backwards after shutdown that little bit, it's because the cylinder is coming up on compression with no spark. They all do that no problem.
Transom. yes. Some of them are stains, the long one from the plug curving up is a crack in the gelcoat. also from the rotten transom. The others on the sides are cracks also.