The problem is:
1) the tank is not cycled. That throws the parametres off, causes them to fluctuate and will almost certainly eventually kill the fish, especially if you don't keep the water changes up (in the case of a fish-in cycle, the changes have to be done pretty much every day). A tank takes on average about 6 weeks to cycle and only once the cycle is done is it safe to add fish.
2) the tank is too small. Paradoxically, the smaller the tank, the more difficult it is to maintain and keep water parametres stable. Nano tanks are usually recommended for more advanced keepers, beginners fare better with bigger aquariums as they are more forgiving and have a bigger margin of error. Things like temperature, ammonia, nitrites can fluctuate wildly in such a small volume of water, and that can severely stress or even kill the fish.
Solutions: Get a bigger tank (5gal minimum, the bigger the better) along with a heater and filter, read about the nitrogen cycle, let the aquarium run without any animals for a few weeks until the bacterial colony establishes itself and the parametres stabilize, then test the water for ammonia, nitrites & nitrates (or ask your lfs to do it for you) and once you get good results you can add fish. I'm sorry but there's no other way around it. Once you get the hang of it it will be a very rewarding experience, but if you try to take shortcuts you will just end up stressing or killing every new animal you get. Patience is your ally, my friend