Your gears are shot. See how the teeth on the ring gear are sharp, they shouldn't be sharp. If you can pull your pinion shaft up and down, that means your pinion bearing and needle bearing are shot, too.
You better get that pinion nut off before you take the diff apart or it will be a lot harder to get off. You won't have any way to hold it. Run your axle back through the diff and use an impact wrench to take it off. Hold the axle with a pipe wrench. They are hard to get off, you will need a good impact wrench.
Most 4-wheeler problems are caused by a loose nut connecting the handlebars and the seat!!
You only need two tools in life -- WD-40 and duct tape. If it doesn't move and should, use the WD-40. If it shouldn't move and does, use the duct tape.