Plants do take some of the ammonia out but they also provide platforms for bacteria to colonise and can be important to remove the nitrAtes from the water. They also put oxygen into the water which in turn helps Aerobic bacteria (those that need oxygen to survive and multiply) to survive. If there isn't enough oxygen in the water your tank will take longer to cycle. I would agree that you don't want too many plants at first but a few help!
Overall its just patience (the hardest bit) some tanks can take up to 6 weeks to cycle, but in the end fish in these fully cycled tanks will live much longer healthy stress free lives!
Don't think your being paranoid! the water quality looks to be the problem, fish condition and colour deteriorates with stress, caused by the levels of ammonia and nitrite in your tank. The ammonia also causes "burns" which could be resulting in the redness on the tail.
A water change every day until ammonia and nitrite read 0 is what you need to do, restrict feeding to avoid uneaten food rotting and contributing to pollution.
If your really concerned get something like an ammonia lock solution from your lfs which turns the ammonia into a temporarily non toxic form. Hope this helps!