The snail needs to get out of there. During the cycling process (which takes anywhere from 4-6 weeks) nitrogen levels with spike dangerously and can drasticaly reduce your fishs' life spans if they are exposed to it. The plants might me okay, but I'm not sure.
I know this feels stressful and it sucks to get the fish and the tank and then learn all this stuff. I'd take the tetra and the snail back, and get a small temporary tank for the betta while your tank cycles. A 5 gallon like the one I listed before will be perfect AND later can be used as a quarantine tank.
For a good way to cycle your tank, read this: http://www.csupomona.edu/~jskoga/Aquariums/Ammonia.html
It is the fastest way and will take advantage of the bacteria you've already introduced.
Once the cycling process is complete you can go get four tetras and put them in the tank, then wait a week before adding the betta or vice versa. You were right to add fish slowly, but tetra are very very stressed when by themselves (aka, without other tetra).
The fish store should take the fish beck, but it is MUCH better to keep the betta as the pet store conditions will likely kill him/her. If they won't take the fish back then you really need the 5-gallon to keep your fish in while the tank cycles.
What is happening sounds like an overload in the biosystem. You have the plants plus the fish and it's all a bit much for the newly cycling aquarium. Tank cycling sounds very complicated, but really it is just creating a balace in the tank- or a mini eco system of checks and balances. Pretty much you are growign good becteria to help regulate the amonia and nitrite, and then 20-30% water changes are used to get rid of some of the nitrate. The aquarium will be more stable without chemical add-ons and you won't have to change the water as frequently which means your fish will be less stressed.
Alternatively: If you really do not want to follow the above advice then what you can try is chemicals. PH is not a big issue so long as it is stable and not too high. Your big factors are amonia, nitrate, and nitrite. There are chemicals to lower those as well.
You could also dump the water out, add fresh tap water, add a conditioner and test the water.
If the levels are good then add your fish back and preform routine water changes. It will take a long time for your tank to cycle this was as the conditioner you use may kill at lot of the bacteria you are trying to grow, but if you stop using the conditioner you will drasticaly reduce the lifespan of your fish- and might even lose them.