I have never heard of slime disease but fish do have a thin layer of mucous over their body and fins, it helps them slip through the water easier. If the fish are stressed or in poor water quality, they produce more mucous and it looks like they have a cream/ white film over their body and fins.
Cleaning the environment up with big daily water changes, gravel cleaning and filter cleaning will normally fix the problem.
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If fish are being attacked by protozoans like Costia, Chilodonella or Trichodina, they get cream, white or grey patches on their body. All protozoan infections cause fish to rub on objects in the tank. White spot and velvet are also protozoans and infected fish will rub on objects in the tank.
The easiest way to check for Velvet is to shine a torch on the fish after the tank lights have been turned off. If the fish have a yellow/ gold sheen to their body, and they are rubbing on objects in the tank, they probably have velvet.
White spot shows up as small white spots/ dots on the fish and it looks a bit like someone has put grains of salt on the fish. Infected fish will rub on objects.
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See if the local pet shop will take the red tail shark off your hands. Most pet shops give a store credit for fish they buy in. Some places give half retail price in credit and others give less. Ask the shop if they give credit and how much they give.
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The platy has a bit of excess mucous and clamped fins in the pictures but seems better in the video. Regular water changes will usually fix that and prevent most other fish health issues from occurring.
The platy might have gill flukes, which can be treated with Praziquantel (available from any pet shop), but that's about it. You can treat them after you do some water changes.
The following link has some information about what to do if your fish gets sick. It is pretty long and boring but worth knowing. I recommend printing it out and reading it in bed to help fall asleep.
http://www.fishforums.net/threads/what-to-do-if-your-fish-gets-sick.450268/#post-3804819
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You never treat fish with chemicals unless you know what is wrong with them. Your fish don't look like they have anything major to worry about.
The following link is about a ram and some catfish but might be of interest to you, especially post #13, which explains when you should treat a fish.
http://www.fishforums.net/threads/help-with-synodontis-petricola-cuckoo-catfish.450894/