Never, ever believe anything a shop worker tells you, always research for yourself. Most shop workers haven't a clue and will make up any rubbish.
Nitrite at 5 ppm is high enough to kill fish, and could be the cause of the betta's death. You don't give an ammonia reading - are you using strips, they don't include ammonia - and that is probably high as well. Both ammonia and nitrite are toxic to fish and must be kept at zero. The way to deal with this is by doing water changes - as many as necessary to get ammonia and nitrite down to zero, then as often as necessary to keep them there.
Feed the fish only a small amount every other day until you get ammonia and nitrite under control. Fish excrete ammonia, and fish poo and uneaten food are broken down to ammonia, and the less food that goes into the tank, the less ammonia is made. In a cycled tank, there is a colony of bacteria which 'eats' ammonia, but they turn it into nitrite. A second species of bacteria 'eats' nitrite and turns it into nitrate. It takes a few weeks for these bacteria to grow enough of them, and this process of growing them is called cycling.
The fact that you have a nitrite reading means that you already have a number of ammonia eating bacteria, but probably not enough yet. And the nitrite eaters also need to grow a lot more of them.
Fin rot is caused by poor water conditions - all that ammonia and nitrite. White spot will also infect stressed fish more easily, and ammonia and nitrite in the water stresses them. Getting those levels right down is very important.
The tank is suitable for the fish you have, but your hardness is soft. This is good for neons and rams but bad for platies which need hard water. Is there any chance of returning them to the shop? Fewer fish will also mean less ammonia made by the fish, and less nitrite made from that ammonia.
Once your ammonia and nitrite levels stay at zero by themselves, you'll probably be thinking of more fish. Please don't get another betta because they are not community fish. They are best kept in a tank on their own.