You can certainly paint the tanks and sides with normal paint -- but you do not want a thick coat of paint on the core (fins) where heat transfer occurs.
As noted, there is a specialty paint for this but I just use spray lacquer and give the core a light dusting.
As for color -- The TRA judging guide (available as a download) says that
"Radiators and securing hardware should be painted black. The brass tap should be unpainted". It does not specify a level of gloss -- but I would describe the original on my TR4 as semi-gloss. Not a shiny high gloss but also definitely not a satin. Then again, mine might have been a high gloss prior to 50 years of cleaning.
Just for the heck of it (as I will never have the car judged) I also painted the tags on the header tank: