Right let's start at the begining
As far as I know popping out of gear is not a symptom of an input bearing and I have driven one until it exploded. Noise is the first sign, followed by judding when releasing the clutch slowly
The geaboxs are very strong and reliable as long as they are maintained, with no work done to them except oil they have passed half a million miles.
Lack of oil will kill the plastic cage on the input shaft, unfortunately checking gearbox oil seals and toping the gearbox oil is often skipped even by garages
People have put over twice the normall torque through them without any problems
Plenty secondhand boxes being sold on ebay with failed bearing and seals, there just up to that age where servicing regularly gets missed.
Here is the drive plate visable through the inspection hole, tourch shining at it, there maybe a rubber cover over it
If you want to know if the input bearing or seal is failing just put your finger inside here (Engine Off obviously) rub it on the inside and see if it comes back covered in oil
Back to popping out of gear
Yes rusty selector cables will do this, I personally had 1st do this but no other gears, Reverse was also sometimes difficult to select, mine was time sensitive, worse when warmed up
The gear selection front to back does not work how you expect it to, it's the momentum of a counterweight that moves it the last bit, any extra friction either jams the system or it does not engage fully
Diagnose
If it's consistent you can manually select first with the engine off and try it, I can just get my arm down on the panda without removing anything, it is uncomfortable, on the 500 you may have to remove the battery tray
Here's me test the gear selection on a panda
If it's intermittently failing, put some very thin oil down the ends of the selector cables and work it in, do not use grease as is often recommended, I attracts the dirt very quickly, see if it improves
I don't know if they ever used the different select mechanism as used on some of the later post 2012 panda 4x4 on the 500
It could be a worn gearbox, which is why you try and diagnose first. I would be very rare as in I have never seen one on a fiat panda or punto