Ok, first of all breathe.
Right.
Fish produce waste, just like any other animal. They poo and wee and do all the rest. In a fish tank, this waste becomes something called
ammonia. This ammonia is very toxic to fish and because they are in a small tank and not a river or lake, they can't get away from it and it builds up in the tank.
In an unfiltered tank, you need to keep very few fish (so in 6.6 gallons, your betta should be the only fish) and you need to do lots and lots of water changes (maybe 4+ a week) to keep the ammonia really, really low. If the ammonia gets high, then the fish get sick.
If you hve a filter, that's great. What a filter does is provide a home for special bacteria that get rid of the ammonia for you. However, in a newly set up tank, the bacteria have not grown yet and so you need to treat it like an unfiltered tank for a couple of months. This means lots of water changes. The best thing to do is get some liquid test kits that test for ammonia and something else called nitrite. Nitrite is also a poison that you get in fish tanks and there are some special bacteria to get rid of that as well. The best way to keep the fish safe is to only have the betta (so that the ammonia doesn't build up too fast) and to test the water every day. If you get a reading for either ammonia or nitrite, do a big water change (about 50% as minimum). This will help keep the fish safe and stop them getting sick from living in their poo and wee. After a while the bacteria will start keeping the fish safe for you (you'll know when that happens because all your water tests will be zero) and then you just need to change the water once a week!
Remember, if you don't have a filter you can't grow the right bacteria and so you will always need to do lots of water changes.
Now, why did your fish get sick?
1) It could have been some ammonia or nitrite in the water
2) The betta could have killed the neons and/or bullied the platy
3) The fish could have been stressed from the change to your new tank
4) The new fish might have come with a disease
Stressed fish = unhealthy fish. They can't take a break or go and watch TV - they find a lot of things stressful because they are still wild in their minds and always watching out for predators and other things that will kill them! They are also always affected by their environment. If we live in smog and smoke we can move away or put the air conditioner on. However, if fish live in nasty water they just get stressed and sick.
Fungusy looking stuff could be fungus but could also be a bacterial condition. Some of those look a lot like a fungal infection. If fish start to produce too much slime, this can also be mistaken for a fungal infection. They produce slime to protect themselves (like a shield) but sometimes they get sick and produce too much. Pimafix is good but some people have reported it hurts bettas (Melafix is the usual culprit, but some people say Pimafix also hurts them). However, the shop was wrong when they said fungus was ick. Ick (or ich) is actually a kind of paraite that looks like grains of sand.
What you need to do now is test your water - go buy a test kit and tell us what the readings are. In the meantime, do a nicwe big water change - I recommend about 70% today and 70% tomorrow. Make sure you add water conditioner (that removes chlorine, chloramine and heavy metals) whenever you add tap water to the tank and make sure you roughly match the temperature of the new water to the old water with your hand.
If you keep the platy in the bowl, keep it in a warm place and alternate between changing 70% of the water one day and 100% of the water the next day. When you change 100%, slowly get him used to the new water by keeping him in some old water and adding a little bit of new water every 5 minutes or so for 20 minutes, and then move him over.
Have a read here - this is all excellent information about problems with fish and how to mantain a fish tank sucessfully!