At least in principal it is indeed possible to process data from a collection of radio telescopes into images, and they often overlay the data on optical images to see regions of radio emission. Whether that's in the reach of amateur equipment is another story that I don't have insight into.
Yes! Radio astronomy at most frequencies is totally feasible 24 hours a day, as long as you’re not pointing too close to the Sun.
Source: I’m a radio astronomer.
It would certainly be cool if it were possible.
Check out the Galaxies tab here: https://public.nrao.edu/gallery/