For corydoras you want at the very least, three of them, as theycare schooling fish.
With a betta and everything I would say at least 20 gallon because cories are dirty fish. Make sure the substrate is either sand or very small smooth gravel. Cories are bottom feeders and like to play with the sand. Larger or sharp substrate could hurt their sensitive barbels.
Corydoras are bottom feeders and will clean up left over food on the bottom, but they DEFINITELY need food other than scraps to fulfill their nutritional needs.
A few days ago I found this thread because I was looking into cories as well: http://www.fishforums.com/forum/catf...-properly.html