No - aircraft weather radar does not show up on an ATC display. It would be highly counterproductive to run ATC radar and WX radar on the same frequency because the scopes (both ends) would be full of annoying lines and squiggles. You can sometime see this if an airplane coming at you has their radar on. If my rusty memory serves, the airplane radar is around 10 GHZ and the ATC radar is around 2-3 GHz
Yes - it is possible with the right gear to find aircraft radar. This would be something any halfway competent military could do.
The other question - When I was working at Westinghouse we were proud of our ASR-9 radar. It has vastly improved ability to show weather and aircraft compared to what came before. Knowing a lot about the project, I used to suggest certain settings to the local controller to help me out with weather
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Radar Basics