He's an omega level mutant (basically so powerful they defy classification). He literally manipulates things at the atomic level, the ice thing is just how he best comprehends his own powers.

We're already 70% water, what's another 30%?

The in-universe science behind Cyclops' optic blasts have been very inconsistent over the years.

The explanation that I'm personally familiar with is that his eyes themselves are portals to a dimension of pure concussive energy that doesn't produce heat, and that energy is also conveniently blocked by ruby lenses in his glasses and visor, and also his own body is immune to it.

Which doesn't really make any sense from a real world physics perspective, but that's comic books for you.

And of course depending on what timeline/reboot/alternate universe you're dealing with, who's writing it, and what's convenient for the plot, any of that can go right out the window, I've definitely seen him melt things and start fires with his blasts in some versions.

I think another explanation that gets used sometimes is that his body absorbs sunlight to power them.

I thought that's what his eyes were, and the lens just reflected the energy back into the "energy dimension"?

Wasn’t it Ruby (or sapphire) back in the day? Might’ve just been the cartoon but I think they had that in there at some point. It’s such a weird power that at I think writers just go with whatever they can to carry the story forward.

One ticket to the laser dimension please

Presumably the same way your body regenerates; his is just faster. And IIRC, he has to eat a lot after sustaining an injury to get what his body needs.