Two things:
First: To update my situation - The new fuel pump solved the problem. Replaced the pump and fuel line, and gas started flowing wonderfully. As of now, the mower is working just as it should.
Travis, as for your situation: You may need a carb kit to essentially rebuild the carburetor. I think they can be purchased for under $10, and just contain gaskets/jets/other small parts.
Modding the solenoid shouldn't create backfiring while the engine is running...at least I don't think it should. When the solenoid isn't functioning properly, you'll get a backfire inside the carb when the engines shuts off and fuel is still flowing into the carb.
The most common reason for backfiring, from what I have learned through reading about this mower, is a problem with the flywheel - mainly the key. You said that it looked normal, but if something is wrong with how that wheel is functioning, you will get the exact same symptoms as you're describing.
Another thing to note regarding this Kohler 17.5 engine: Buy a more powerful battery than you think you'll need. Originally, the previous owner of this mower tried to get away with a battery rated for a 14hp engine. It had a horrible time cranking. I just dropped in a battery rated for 19hp engines, and it cranks just fine, but might even do better on an even bigger lawn battery.