It is normal to have debris after the water change when you put a lot of water into the aquarium very quickly.
First, you need to make sure all the organic debris are removed during the water change. That means all the fish poo and leftover fish food and any other waste product in the aquarium that can rot. So what left is only the substrate. Most substrate will not floating around for long and should settle within seconds.
If you mean the substrate was floating around all the places, you can reduce it from happening by putting a small plate in the tank and have the new water drop on it when you fill the tank back up.
I am not what to make of the grease-like substance. Maybe you should reduce feeding. A betta fish does not need to eat much at all. Extra food means more organic waste in the fish tank.
Carbon pad should be black in color, not white. If it is white, it is just the biological filtration pad and you are not supposed to replace that unless it is falling apart. If you replace it every month, then you are just reset your aquarium back to before the nitrogen cycle ever started for the first time. Ammonia and nitrite can build up until it is cycled again which can take a month or more without heavy seeding.
You can reduce the filter current by install a piece of pre-filter foam or sponge on the intake. It will slow down the water flow. A little current will actually make fish more healthy in the long run. Similar to how humans should stay more active to be healthy. There are reports saying betta lived up to 7~10 years when they provided periodically water current for it to "workout".
