What type and brand of fish food are you using? If you are using pellets make sure you have the smallest ones, the neons would likely have some trouble with anything over 1mm in diameter; if flakes ignore this info.
How much fishfood do you feed them with? Although it seems to be a good idea to let the fish fasten for a day or maybe two per week, feeding two times each seven days is likely to be health threatening for them.
If your readings are actually the ones in your aquarium info then you are still cycling (
ammonia present at 0.5ppm and a spike in nitrites indicates you are in the second phase of the
nitrogen cycle).
Consider daily water changes (close to 50%), and using a water conditioner that locks ammonia and nitrites within non-lethal levels -the only safe level is zero- but that won't prevent nitrifying bacteria to colonize your tank, such as Seachem's Prime, should help your crew survive.
What are your nitrates readings? I guess still close to zero, but once you see them approaching 10-25ppm, your cycle will be ending.
Pepe
Santo Domingo