How often are new astronomical objects (variable stars, supernovae, comets, etc) discovered by amateurs? Where could one report new findings?
The answer is: frequently. There are many amateur astronomers that make it their ambition to discover new supernovae or to observe and report on new variable stars.
As an example, let me cite amateurs Robert Evans, who apparently holds the record for most supernovae found by visual observation, or Tom Boles, who holds the record for supernova discoveries by an individual.
Observations of variable stars can be reported to the Information Bulletin of Variable Stars (IBVS). Supernovae discovery or new comets would normally be reported to the International Astronomical Union Circulars.