Can a photodiode emit light?
Yes it can, and does.  Any recombination of the carriers (electron and holes) which happens all the time will emit light with an energy level corresponding to the bandgap, 1.12 eV (minus just a little bit) ~ 1 um for Silicon.  Although, it must be noted that this is a very inefficient process.  It is not used directly because of that but is used in some diagnostic tools.