Bottom line; the robin is a carbureted engine, and it is showing the symptoms of a common problem eventually shared by all small carbureted engines: stuck or clogged float valve, or waterlogged float. If it was a Kohler, chances of this happening are the same. After you run enough gas through the engine, eventually you might have to clean the carburetor, especially if the engine sits for any length of time idle with the carburetor bowl filled with gasoline.
The float (part 9) controls the needle valve (part 8) to keep the level of fuel in the bowl constant. This "bowl" provides a constant source of fuel to the main jet, through which the fuel is sucked into the incoming air.
If this valve sticks open, the bowl will quickly overflow through the air vent tube, putting a puddle of gas into your carburetor (and in a gravity filled system like this, it could result in putting gasoline into the crankcase, like you reported initially). The fuel solenoid just stops fuel from leaving the fuel bowl and entering the carburetor intake through the main jet when the engine is turned off. With a sticking jet, there is still a path for fuel flow into the carburetor even with the solenoid turned off. With a stuck jet, when you do get the engine started, it might act like it is running rich initially, but when you open the throttle, the bowl will quickly empty, starving the engine of fuel and causing it to stall.
One suggestion is to tap on the side of the carburetor sharply with a wrench next to the fuel inlet tube. You may here a "tink" when the valve comes unstuck, although this will not permanently solve the problem.
Carburetor rebuilds are not actually that hard. All that means is disassembly and cleaning and replacing gaskets and the needle valve. Maybe an hour of work (if you end up with some gaskets that are difficult to scrape off of their sealing surfaces). You don't have to (want to) do it bent over your engine... two bolts and the carburetors off and you can do the fiddly stuff at the workbench (you could also just pop the carburator off and drop it off at a small engine repair shop for rebuild if getting the whole tractor there is too difficult).
The robin service manual doesn't walk you through the rebuild, but there are dozens of generic small engine repair manuals that would cover this. I haven't rebuilt my robin, but I have done this on a number of outboard engines and lawnmowers. YMMV.
You have probably spent more money so far on parts then a small engine shop would charge for a carburetor rebuild. That is the irony of troubleshooting; you always spend more money on finding out what the problem is rather then actually repairing it.
If you want to avoid this sort of problem, then you need a fuel injected engine (or a diesel). Of course, that adds to an exponential increase in engine complexity, and 1000 hours later you will probably be dealing with electronics based starting problems.
Pictures from the Robin service manual:
http://www.robinamerica.com/media/manuals/128937872760282500.pdf
Instead of using a propane torch to dry out your engine after flooding with gas, I would just take the sparkplugs out of the engine and leave them on the bench overnight to dry out and allow fuel vapors to leave the cylinders.
