Hi Kaley,
firstly, can you please refrain from using terms like "spaz out" - it's offensive to disabled people. Thanks.
Secondly, to answer your questions, I think the issue is slightly more fundamental than will the racing around kill your neons. I suspect a much more serious problem (although I'm praying I'm wrong). Did you just buy the tank, set it all up and put the fish in, just like that? If so, you really need to understand about the nitrogen cycle.
Forgive me if you know this already, but it goes like this: fish waste turns into ammonia. Ammonia is deadly toxic to fish, but thankfully there are bacteria which can turn that deadly ammonia into slightly less toxic nitrite. Nitrite isn't good for fish either, but then you have some other bacteria that come along and turn it into nitrate. Nitrate isn't toxic in normal concentrations and is either used up by your live plants, or if you haven't got live plants, it's removed when you do a water change. This is known as the Nitrogen Cycle, and a tank that has the Nitrogen Cycle operating in it is said to be "cycled".
When you first set up a tank it doesn't have any of the beneficial bacteria, so the levels of ammonia rise and rise and your fish die, more often than not. After a few days, the bacteria have grown sufficiently to break down this ammonia into nitrite and the ammonia levels fall to zero - but then the nitrite levels rise and rise. Any fish that survived the ammonia "spike" now have to run the gauntlet of the nitrite spike. After a few days or weeks, the bacteria develop that can break down the nitrite to nitrate. Nitrite levels fall to zero and your fish can now swim happily. Except that you might not have any fish left by this point and those who have survived have been damaged
Fortunately, danios are tough. Neons are less so, however, and they probably won't make it. You can increase their chances by doing a 10-20% water change every day and only feeding your fish sparingly, every other day, testing regularly for ammonia and nitrite to see how the cycling is going. Fish are cold blooded so need a lot less food than warm blooded creatures - fasting is good for them and also cuts down on the waste they produce. I strongly suspect you are over-feeding your fish, which will kill them, sadly
You can speed up the cycling process by getting some old filter media from someone else's filter, plus a handful of their gravel. If you keep the filter media and gravel wet and oxygenated, it should contain the bacteria you need to get things going. I am personally sceptical about products purporting to contain these bacteria already, such as "Cycle" - I think they're a waste of money.
I think your actual mix of fish isn't bad - danios are very lively fish, and neons are a little more sedate but I don't anticipate any problems. However, high ammonia and high nitrite can cause "darting" behaviour, which may be what you are witnessing.
Good luck, and stick around - it'll work out in the end.