Ill put some thought into this.
Where is the fish tank located? Is it in any direct source of sunlight?
Sunlight is the number one cause for algae.
Also, Do you leave your lights on all the time? Make sure it is turned off at nights.
This is also a big problem that people tend to do, and dont know that it causes algae growth.
Those two question should be the first question you ask yourself when you have excessive amounts of algae growth.
Having plants in the fish tank will actually help with the algae growth, as the plant will abosrb the light emitted from the sun or other light source that causes algae. Also be aware that having plants in when your tank isnt cycled yet isnt such a smart idea. It will disrupt the cycle. Plants will take what good bacteria that is needed in the tank to cycle, thus messing up a few things.
ammonia > nitrite > nitrate
If any of those are missing it will disrupt the cycle.
As far as the white parts that are growing on your fish, it could be nothing more than just an excessive amount of slime. Sometimes when the water quality is bad, fish will tend to develope excessive slime, to protect themselves. Sometimes when brown algae appears it'll be because of bad water. Now just because you test the water with a test kit and it reads good, doesnt exactly mean that you are fine. Other factors are involved too.
How often do you do water changes? How do you do the water changes and how much do you take out?
Also from my experience the type of filter you have shouldnt have any affect on algae growth. So I wouldt worry about the type of filter you have.
A general rule for getting the right filtration is usally that it needs the filter the tank 4 times in one hour. The more the merrier.
On cycling your fish tank, you dont really need a source of ammonia. It will do it naturally. Dont add any food, ammonia, or feeder fish to try to speed up the process. Let it do it by itself. A 20 gallon standing by itself should only take 2-3 weeks to cycle.
Also, The one inch rule is crap. Don't listen to that. If you had a 20 inch goldfish you wouldnt want to put the goldfish in a 20 gallon would you?
If you cant keep the ammonia, nitrite and nitrate levels at 0 without doing any waters changes for a while then that means your fish are way too big for the size tank you have. You're fish is producing too much waste and there isnt enough volume of water to hold enough beneficial bacteria to destroy the bad bacteria. Get a big tank if you are having problems with the levels going up.
Both the size of the tank, and the volume of the water affects the quality of the water and the fish.
There is still a lot more I can say. But lets start from here first.
