Just to throw in my two cents: in my experience, neon tetras are never nippy. Actually, they're just about the wussiest fish I can think of. However, they don't make good betta tankmates because they're just so small that a big betta can eat them. Black neons are a completely different species of tetra and get considerably larger than neon tetras, thus making them inedible (for a betta) and thus acceptable tankmates.
Also, in my experience danios aren't nippy at all. They chase each other around a good bit, but I had mine housed with a male crowtail for a while and they never even acknowledged his presence. I have five danios, which isn't so many that you'd 100% eliminate extra-species nipping issues if there were any in the first place. Also, danios are lightning quick, so you don't have to worry about a betta nipping on the fins of long-finned danios. Slower moving fish with long fins like guppies might be a problem, but I'd like to see a betta try to catch a danio.
Cories are good tankmates for bettas because they stay on the lower levels of the tank and aren't even remotely aggressive/nippy. Also, they're armored catfish, so if it turns out that you've got a jerk betta, he won't likely be able to do any damage to your cories even if he does attack them. I would say panda cories are average-sized cories. $8.09 a piece is a bit on the high side, but panda prices vary a lot. I've seen them for as much as maybe $9 a piece but as cheap as $4, so I'd do some more shopping.
Also, there's no magical number of fish that's going to cause you to overstock your tank. Honestly, you can cram a lot of fish into a 10g tank and enough bacteria will grow to handle all of the waste. However, the end result is that fish waste is going to be processed into nitrate, which builds up in your tank. The amount of fish you can put in there is restricted by how quickly this nitrate builds up. So, if you've got a betta and five cories, adding one more cory isn't going to suddenly make your tank go crazy - it will just add a little more nitrate per unit time to the tank, so you'll have to do water changes a little more often or slightly bigger changes.
About breeding: many corydoras species are "easy" to breed. That is, it's easy to get them to lay eggs and fertilize them. However, cories are really crappy parents. They'll eat their own eggs if they can (and they will certainly try), and they'll defintely eat little baby cories swimming around on your substrate. If they don't, the betta will for sure. If you want to breed them, you'll have to do it in another tank or remove the eggs to another tank and raise the fry until they're big enough to survive with the adults. So, don't expect to put some cories in a tank and have their population increase. If you want six cories, start with six from the start (although you'll probably want to add three at a time just to avoid overloading your biofilter).
Hope that helped!
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