I'm curious about one thing. Who tells an employee to put air in the tire of a personal vehicle let alone change it?

Where in your job description is that clause? Sounds like a trip to personel, to file a complaint. Your supervisor is trying to get personal work done for her by an employee on company time.

If one of my supervisors told me that, I'd tell him or her to get out a copy of the contract and show me where that clause is listed.
Then if it persisited, I'd go injured and have my delegate explain to the administration why.