"Is each ATC Tracon or Center equipped with weather radar capabilities? Can GA airplanes not equipped with radar depend on this if they don't have other onboard weather capabilities?" - Steve B.
“First of all, your goal and planning should be never to get yourself in a situation where you have to rely on ATC weather radar to negotiate severe weather. Consider that a Part 135 commercial operator flying a two-pilot, turbine powered airplane is prohibited from flying in night IMC without weather radar.
Onboard satellite systems should be considered a strategic tool only. The update and resolution is not good enough to tactically navigate an area of convective activity. Even onboard radar systems in light aircraft are generally pretty low power and limited range.
ATC radar is primarily designed for control and separation of traffic. Weather avoidance is a secondary function.
Terminal Radar (approach controls) has a normal range of 50 nm where Enroute Radars have a longer range (typically 200nm) and can mosaic even further.
But, even though the terminal radars have a quicker update than the enroute, neither is designed for weather penetration. In times of heavy weather, the controller may have to turn down or even turn off the weather returns just to see aircraft returns – so you can’t really rely on it for weather avoidance.”