They actively care about (as in, act on in realtime) serious criminal activity. But I suspect they keep the legal yet embarrassing stuff in their back pocket for blackmail purposes.
the FBI acts on and handles cybercrime and/or regular crime.
the NSA can absolutely come up with tools that they then share with other parts of the USG, in the same way that the Army can develop the M-16 and allow police departments to use it, but ultimately it's law enforcement pulling the trigger.