No, the cone shell is probably a Malaysian trumpet snail. It could've hitchhiked in on some live plants if you have them, or even with the fish, especially if you dumped the bag of water containing the pet store water into your tank. Malaysian trumpet snails aren't bad, and they generally hide in the substrate during the day and come out when the lights are out, so that would explain why you haven't seen it.
Fish having spasms sounds like some kind of poisoning or irritant, and especially with them dying off so quick as well. Are you using a dechlorinator to remove chlorine/chloramines from your tap water? Did you do a water change and then notice the symptoms? I would suggest checking your ammonia and nitrite levels as well, make sure both are reading 0 ppm. Those can also irritate the fish and cause them to have spasms as you describe.
Here's a link also, measure your tank length, width, and height and compare it to the chart to see what size your tank is exactly.
http://www.cichlid-forum.com/articles/aquarium_sizes.php
As you can see, a 3 foot long tank can be a 30 gallon long, a 40 gallon breeder tank, or a 50 gallon.