Nitrogen cycle not working

I'll start with the history before I get to the question.
A relative perchased a 1.2 gallon tank for my 2 year old niece and put a male betta and an African dwarf frog in it. Knew nothing of the nitrogen cycle or even basic general care of the pets. Both betta and frog got sick and I started doing research and eventually they got a 5.5 gallon tank and put the fish and frog in that. The frog developed dropsy and so I quarantined it to the 1.2 gallon tank where it died. Now the fish is solely in my care. I was able to do a fish less cycle in the small tank and have monitored daily the 5.5 gallon tank. Then another relative got three neon tetras and three zebra danios per the advice of a petsmart employee. The fish were added into the 1.2 gallon tank to be quarantined. But the ammonia went crazy at the beginning of the second week so I acclimated all the fish in the 5.5 gallon tank and they are doing great and all get along great. They even "piggy back ride" on the betta sometimes. But the tank had not finished cycling. I added the filter and plants from the established tank hoping it would help. I have a ten gallon tank filter on the tank as well as a heater. There has been an occasional hint of nitrites and nitrates hold at 5.0. Ammonia will spike to 2.0 sometimes but holds at 1.0 mostly. I do daily water changes to try to keep Ammonia down as much as possible. I know the tank is overstocked but this is the best I can do for them until I can get a larger tank which won't be a while. Now to my questions:
Did I make a mistake by adding the filter and plants?
Are the Nitrates eating the nitrites before they are able to grow enough to eat the ammonia?
How can I help the nitrites grow?
I've been using a bacteria supplement, is that adding to the problem? I cycled the small tank without it but was told it would cycle the larger tank faster and be safer for the fish. Been trying to cycle the tank for over a month.
I'm sorry if all my questions could be answered in other posts, I just couldn't find any. Don't know where to look. Any help will be greatly appreciated!

 

Well first off the neon tetras and zebra danios are too active for even a ten gallon. The neon tetras are also not temp. compatible with the betta. Both are schooling fish too, and would need at least six of the same type. I would rehome them since you don't know when you are able to get a bigger tank. This is probably also why your ammonia had a spike due to the overstocking.

Adding plants and the filter from an already cycled tank was not a bad thing.

The order goes ammonia, nitrites, then nitrates not the other way around. Have you tested your tap to see if you have ammonia or nitrites in it?

Ammonia should be 0, Nitrites should be 0 and nitrates should be less then 20. What are you using to test the water with?

What bacteria supplement are you using?
Are you using a conditioner such as prime too?

 

I do know the order the cycle goes; ammonia, nitrites, nitrates. My concern is that the nitrites haven't grown and I'm worried the nitrates are eating them before they can eat the ammonia. My tank had just started having nitrites when I added the stuff from the other tank which immediately showed nitrates and I haven't seen anymore nitrites since then.
When I first tested my tap water there were none of ammonia or nitrates. But that was months ago, I will retest tomorrow.
I'm testing with API freshwater master kit.
I'm using Top Fin Bacteria Supplement and using stress coat with dechlorinator.

 

Ammonia gets converted into Nitrites. Nitrates then get converted into Nitrates.

It seems like the nitrite consuming bacteria is doing its job and is populating fine.

Your ammonia consuming bacteria seems to be struggling with the 7 fish in the 5.5. How long has the ammonia been high?

 

I think part of your confusion can be put at ease by clarifying that Ammonia, Nitrite and Nitrate are all chemical by products of Living organisms. In an established aquarium Ammonia is produced by fish that are just kind of existing. Digesting food and even I'm pretty sure even just water running over their gills can produce a small amount of Ammonia. Ammonia is eaten by some specific beneficial bacteria and the by product of the ammonia being eaten is Nitrite (another Chemical even more toxic than Ammonia) The Nitrites are then eaten by another bacteria and the by-product of this is Nitrate, (yet another chemical which is less toxic is low levels and even necessary for growing plants) I think, And I could be wrong as I'm not sure I fully grasp the finer details, that its best to keep Nitrate levels under 20 ppm in an established tank via water changes and live plants that use up the nitrate to grow.

When Cycling a tank the goal is to foster an environment in which all three chemical producing organisms thrive. The bacteria that eat ammonia do well because there is ammonia to eat, the bacteria that eat the Nitrites do well because the ammonia eating bacteria are producing lots of nitrites to eat, and the Nitrate level increases because the Nitrite producing bacteria are doing well. When the Ammonia and Nitrite eating bacteria are doing well they naturally remove the toxins from the water to make it safe for the fish to live and the fish (just by living) will produce more ammonia and keep the cycle going.

Your fear of the Nitrates eating the Nitrites before they have a chance to grow should then be put to ease because Nitrite and Nitrate are chemical compounds and by-products of bacteria so not even being alive, they can't eat each other.

Ammonia levels should be at 0.0 ppm And Nitrite levels should be at 0.0 ppm because all of the ammonia and Nitrites being produced are being eaten by bacteria which is really what you're shooting for. If you say that you're seeing a lot of a nitrates and not a lot nitrites or ammonia then that probably means that your tank is cycled already and in order to keep the nitrate level down you should be doing regular water changes to keep the ecosystem happy and healthy!