Your local lawn mower shop may have an air filter that will clamp on to the carb. without one any dust that gets in the motor will wear out your rings faster than normal.
use a volt meter to test the rear light bulb, or a test light will work too. All 3 of the wires for the tail light should have continuity between them (connection) so disconnect the wires and get the tail light separate, then check with the bulb installed, no matter how you test, there should be continuity.
Also check the main leads that follow under the tank, i believe the brown wire should your positive lead. The solid green is ground. If you have black wires, those are for the turn signals, dont worry about those for now.
The rust in your tank will always slowly break down, ive attemped to clean mine 2 or 3 times, but still more comes out, hence the importance of an inline filter. If you remove the petcock, a whole bunch of junk may fall out, this is good, clean out as much as you can for now.
let us know if theres anything else we can help with.