If the starter solenoid is clicking then that means you're getting voltage from the battery, through the harness, to the starter button, back through the harness and into the solenoid. Now you just need voltage across the solenoid.

To see if the solenoid is bad, jump the two big posts on it and see if the starter engages. If it does, then the solenoid is bad. If it does not, then that means that the big wire leading out the solenoid isn't sending power down to the starter, or the starter is bad. You can test the starter by putting jumper cables on it. The hot lead on the post and the ground lead to anything metal on the chassis or engine.

As far as the engine covers,,, yea they are needed. The LT80 is kind of like a weedeater engine. Has a little fan that's powered by the crank that blows air up and over the cylinder and head.