Doctors use a thing with a little spinning wheel (size of flint wheel on lighter) on the end to flick imbedded debris off your eyeball…been there done it. A experienced doctor will prescribe eyedrops for a few days prior to removal, that will help loosen the debris. This is from experience..
And what country did this happen in ? ?
Not a good idea, if you have the magnet the wrong way ie, the pole the wrong way you can drag it across your eye.
I've had it done a couple of times (slow learner) works good but scary lol, they put eye drops in to freeze it but when they do it the surface of your eye jiggles like water lol.
Getting a stick with a magnet near my eye would be weird. 
You don't have any cow magnets lying around?
Can't find the old solid polished type these days. Now cow magnets are stacked multiple magnets on a plastic frame. But the one I used for my eyes was way more powerful than the old style cow magnets, but you are right, the old style would work fine as they couldn't have been polished smoother!
Just looked them up and the old style are still available. Guess because new style are cheaper, my sources all went to them.