OK, did the mower thing....Yes, Jimmy can jump on top of the mower..and shoot.
Problem is....with the exception of the Hattrick mission, the only access you get to the mower is dentenion and the Park and Yard jobs...In all cases, no one else is around to fire on except the Prefect at the Football field, and the "Boss Guy" on the paying mowing jobs....You can shoot these, but your mowing mission is failed at the first shot....Second problem is....Jimmy has only 8 seconds to 'get back on the mower'...and that doesn't mean standing on it....Third, is you have to line Jimmy up exactly in the front of the mower, with your screen view aligned, and he has to jump, unlike the Go-Cart where you can 'walk' him up from the front.
I believe that after one of the Hatrick missions, I was able to drive the mower away...But it has been too long, and I would would have to do that particular mission again to be sure....If any of you have a chapter saved right in front of that mission (I don't), you could check this out and see if Jimmy could indeed drive the mower around town for the rest of the day.....