Question 1: Have electrostatic discharges within dust storms been detected and quantified somehow on Mars using standard radio communications equipment?
Not "standard" equipment (DSS-13, the 34m radio telescope (image)).
The evidence we have so far (1), which was only detected briefly and has not been observed again (2,3), used fancy processing of the ratio of several variables in the radio signal to attempt to differentiate lightning from background thermal radiation.
Question 2: Have there also been dedicated receivers or lightning detectors on Mars for this purpose?
Yes and no.
In terms of detecting lightning, there are a few options: you can detect it optically, acoustically, electrically, or by radio emissions (4). These sensors can be on the planet, in orbit, or on Earth. Mars has a few challenges with some of these methods, though: the dust storms that would create lightning block any light emitted, and the atmosphere doesn't conduct sound well enough for a local sensor to have much range. Thus most of the efforts have focused on detecting either the charge field (electrically) or on capturing the radio emissions from a lightning strike (by radio). True confirmation of lightning will likely have to come from at least two different methods (4).
Initial experiments focused on using radio telescopes to listen for lightning on Mars, but only one signature was ever detected, leading that result into question (2). Mars Express, which arrived at Mars orbit in 2005, had a radio antenna which was used to listen for lightning strikes, but it did not detect anything over a five-year period (5,3). This used the background from the MARSIS instrument, so it probably wouldn't be considered "dedicated". However, the Schiaparelli lander, part of the ExoMars mission, contained the MicroARES instrument to measure the electric field generated in Martian dust storms with an antenna (which would provide evidence for lightning), but it failed to land safely (6,7). This would have been the first dedicated instrument for lightning detection to land on Mars (4). MicroARES would have also been able to detect if electrical discharges were just "bleeding" back into the atmosphere, without actual strikes.
(Image with MicroARES circled in red, modified from the original by Gerbil at Wikipedia)
Since the Schiaparelli lander, it appears no new instruments have arisen, although new research suggests that lightning may be exceedingly rare on Mars due to the atmospheric pressure, giving the 2006 measurement some more leeway (8).
