Tons of people keep betta and cories together, myself included. I have never had any problems with them together. Even a slightly more aggressive betta will usually accept them pretty soon simply because they are dull and don't get in the betta's way.
I agree, you should really up your cory group to 5-6 cories, and maybe some shrimp or a snail. Cories need groups of 4 or more to feel secure.
Quarantining is done with new fish from stores, just in case a fish brings in a disease, you can watch for it in a separate tank instead of potentially introducing it into your established tank.