Man, sounds like you found all sorts of issues. A bad water pump can lead to a lot of issues. Wonder how much just fixing that will help your performance. Also sounds like you need a bottle of SeaFoam to clean out your pistons and such. Guess the previous owner was running low end fuel and didn't spend the extra money for high octane
I took it apart to preventively change the impeller and gaskets, since they were cheap enough. Glad I did...
The PO used dirt cheap TC-W3 oil, and I suspect that may be part of it. Someone pointed out that there really isn't ongoing quality checks to ensure the oil continues to meet TC-W3 standards, so the companies could start taking shortcuts, or just not exercise much QC. I'm switching to the Pennzoil XLF synthetic blend 2-stroke oil as the reviews are 99.999% positive, citing lower smoke output, better combustion, etc. I'm using up the last of the crap mix for basic testing, and will tune the carbs on the new stuff. I'm using 91 octane, ethanol free, non-oxy fuel.. My local gas station sells it, so I'm happy about that. I have sprayed seafoam into the carbs to clean out the engine, but now I am thinking I am going to get two spray cans, fire up the motor and rev it up in idle to full, then spray the entire contents of the cans, one into each carb, then shut it down to let it "hot soak" after the cans are empty to really clean it out. Then when I get the carbs adjusted, I'll drop in an entire can of Seafoam into a tank of gas and just spend the day running at WOT as much as possible to run it through. Afterwards I think the inside of the engine should nice and clean.
I had initially taken out the spark plugs, sprayed Seafoam in the cylinders, put the plugs back in and let it sit for a couple days. It fired up a lot easier after that for awhile, then got harder to start later. I'm really thinking that if it isn't the carb adjustment, then the cheap oil/gas mix isn't fully burning up, leaving crap in the cylinder and on the plugs to gum up. I noticed a spritz of Seafoam into each carb before starting it cold lets it fire right up, time after time. It runs great if starting off Seafoam, but that's not practical to do every time it's cold...
The symptoms mostly point to it running lean either from carbs being out of adjustment, or fuel delivery issues. I am rebuilding the fuel pump preemptively, and I might need to replace the bulb in my fuel line, I think the check valve isn't working right, so fuel may be flowing out of the motor back into the tank, and creating air bubbles in the lines.