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1. save a "start" point when the player steps onto ice and return the player there if they fall into the cold water
Breaks if there's an island anywhere. Step on island which is not ice. Burn all ice around island.
"Make it so they can't burn ice while standing on land."
And so the quest begins...
Unless non destructible areas are set by you and not actually whether or not a tile is destructible. In which case, an island would be marked destructible. Or just have no islands surrounded by ice.

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2. have the ice sink if the player is standing on a single square
This is good, as long as the lighter is infinite. If not, you may run out before you can make yourself stand on just one block.
But if walking in the water will eventually kill you, and killing you sends you someplace safe I see no reason to even do this. Forcing the player to kill themselves works, and maybe they'll learn something.

Making a single square of ice sink is basically forcing the player to kill themselves, anyway. (They have to burn all ice around to grant themselves death.)
Cases like this is what makes game design "fun". If it were me, I'm not sure how hard I'd try to protect a player TRYING to get themselves stuck. Golden Sun will start you at a safe point instead of your last position after loading the save if you hold a button combination while loading. Your game puts you at a safe place when loading automatically. With a fail safe like that, I'd do whatever and worry less.
It's even possible to get stuck in "simpler" games (as far as fancy things the player can do) like Pokemon. I don't see it as bad design, if someone has to go far out of their way to make it happen.