Yes, assassin snails prey on all snail species, themselves included. They reproduce slowly, and once they eat all the snails, you often end up with only one or two left.
And why would you need fire to kill snail eggs ? Just scrape them off and toss them out, they dry out quickly. Live snails, well, you can crush them, and some carnivorous fish will eat the meat without the shell and some don't.
The eggs you picture are from ramshorns, not the little bladder snails. Bladder snails lay an irregular blob, ramshorns lay that neat little crescent of eggs. Many call bladder snails pond snails, and you do find them in ponds, but the pond snail itself is a larger species and not that common in tanks.
Bladder snails are super common and reproduce quite quickly, laying eggs on anything at all, from a young age. They and the ramshorns are hermaphrodites, so don't need a mate, but they prefer to mate if they can, it enhances their genetic pool.
Mystery snails, aka Inca snail, are male/female and lay eggs out of the water, in a big pinky beige line about a half inch thick. They take weeks to hatch, and if you don't want them, just toss the egg case, it will dry out too. If these snails are trying to spawn they can climb out and fall off, which is fatal if you don't find them soon.
Females can lay many batches of eggs from one mating, so even if you only have one, if it's female it may lay eggs the first year, if it has mated before you got it. There is a giant ramshorn that is related to the mystery snail, really quite neat, and even more inclined to climb out of the tank and fall off.
Some loaches and puffers eat snails as a regular diet. The loaches are sometimes kept to control snails.
Don't use meds to kill snails, it will kill all inverts. Don't overfeed, and hand pick any you see and that keeps numbers down. It's kind of a personal thing if you want them or not, but they do a fine job of cleaning up leftovers,dead plants, algae and cleaning glass too. I like having some of them around, but prefer ramshorns to the bladder snails, especially the red and blue coloured ramshorns.
