I couldn't tell you exactly what the solenoid does but I think it is there to prevent backfires. As you have seen it does nothing to help keep fuel from leaking past your carbs inlet needle and into the intake manifold.

What you do need is a manual fuel shut off valve, like this:

http://www.homedepot.com/p/Unbranded-3-25-in-W-x-1-25-in-H-Fuel-Shut-Off-Valve-5091/202386208#.UqCkueqEiic

10 minutes to install. Put it between your gas tank and fuel filter. It will stop the leak while the mower is not being used.

The optimum fix is to fix the carb inlet needle or float, but you may not need to and one should install one of these valves anyways, so I would try it first.

Excellent advice.
My neighbor had the same problem on his Toro tractor.
He had to keep changing the oil and it became quite costly.
Now after he uses it it he just shuts the fuel line closed and no more gas in the oil. The carb had been rebuilt once before but the issue appeared again. The fuel shut off cured the issue permanently.

Question: What is the purpose of the solenoid on the bottom of the carburetor bowl?

electric operated fuel shut-off to the main jet to prevent backfire or loud pop after immediately shutting down hot engine.

You can audible hear the fuel shut-off valve when you turn the key on you will hear click at the carb. bowl. No click the solenoid maybe stuck or defective. Can be cleaned and tested by applying 12V to the single wire and grounding the other to the case of the solenoid. Should see it pop open and closed (up and down) when removed.

Fuel getting into the oil is from the float needle not seating and leaking fuel, beings the carb. is above the crankcase gravity takes it's course and fills up the crankcase as described above.

Thank you people!

I had already thought about putting in a block valve. Strange it has never happen in 14 years we had it. And this summer it once set for 4 months! My ATV came with a gas shut off that I always use. We don't live there, only go there maybe once a month or so.
I priced the carb kit online... wow it's high! I'd rather not spend that much so will go with a block valve. Mower is at our farm and we have the farm for sale so only need it to last till it sells.