Hello and welcome to the forum.
Lets talk about the oil first. Some water supplies (mine is one of them) are prone to a film on top. I still haven't found out what causes it but my water does the same thing. I can leave it standing in a bucket for a few days and get the same thing so I know it isn't anything in my tank. I will say however, that you can also get a film if you put your hands in the tank with oil or grease or even hand lotion on them. There a couple ways to keep in off. You have found one already but it is time consuming and never really worked well for me. If you have a HOB filter, you can lower the water level slightly so that the waer splashes back into the tank. That will keep the film broken up. Or you can add airstones to the tank. The air bubbles breaking the surface should also keep it broken. That is the option I use.
Now lets get to the more important matter of your fish. Since you have had your tank for a while, we can probably rule out in not being cycled but I will ask if you have checked your ammonia and nitrite levels to make sure they are 0? It's possible that ading the tetras last week cold have caused an ammonia and/or nitrite spike if you only had the betta in the tank before. How large is the tank and what fish were in the tank before you added the tetras?
If it was a pet store betta, 2 years is a fairly typical life span for them. As they get old, they usually do exactly what you described, laying in one place on the bottom or under a plant for a few weeks until they pass on. They also seem to have some swim bladder issues and can't stay upright or have trouble getting off the bottom to go to the top of the tank.
Having said what I did about the water and film earlier, since you only started seeing it recently, it could well be that you have gotten something in the tank that has caused it and subsequently is killing the fish. If so, the betta would be particularly at risk since he is a surface breather and would be picking up whatever it is everytime he goes to the top to breath.
Also, how did you acclimate the new fish? It could be that the pH/GH/KH of the fish store ater was considerably different from your tank and the new fish suffered from such a sudden change.
Sorry for all the questions but the more information, the easier it is to find a solution to the problem.