1) I only take out my rear axle if absolutely necessary. Repeatedly removing and replacing that axle will cause wear on that plastic. I know others have reported this to you too, but I just don't see there being much wear just from normal use. I think people have fixed this by applying small amounts of epoxy in the axle mounts, the redrilling the holes.
2)New battery? Most people yank that Tomy battery soon after buying a new bit. Do you doublecharge?(of course I would suggest replacing controller batteries also)
3) Why would you want to control the car while it is charging? All you are doing is wasting your charge.