Radiation does not by definition make things it impacts radioactive, and most of the lingering effects that we associate with radiation have to do with radioactive particles that are left behind by the atmosphere. Since there is no particulate matter traveling between the sun and Mars (photons not withstanding), the surface of Mars would not be expected to acquire radioactive properties from an acute solar radiation event like this one. However, an entity on the surface of Mars would be exposed to radiation from the Sun, much more so than an entity on/near earth and it's magnetosphere (Mars' thin atmosphere and distance from the sun probably helps reduce that disparity relative to something in say low earth orbit, but I don't think it fully equalizes or shifts the scales). I am not an astrophysicist, so there's at least a 20% chance I got part of that wrong.

Yes, and radiation shielding is a major challenge for humans on the mars surface.