Neons are usually mid water range fish, and cories stay on the bottom generally. 10 neons seems abit much for a 10 gallon. Perhaps 8? I'm saying that because my 5 neons are right piggies and have gotten bigger than any I've ever seen in a store. Buncha oinkers! That being said, with mid range and bottom feeders you could possibly add a couple of surface fish without overloading the tank, but if you're new to fish keeping start out slowly and then work the numbers up. You're right, tetras love company, keeping too few makes them nervous. So what the hey, have a neon tank! Cories seem to get along with everyone, they're like little cows grazing along. I have neons, glowlights and black phantom tetras in my larger tank and they love all schooling together in the mid range and what a colourful bunch they are. Go go tetras!
I think that with proper attention to water changes, 8 neons, 3 of the smaller cories and 2 dwarf gouramis would be fine. Neons and cories both produce very little bioload for their size so I don't think that adding in 2 dwarf gouramis would overtax the filtration if proper attention were paid to timely water changes.