I was trying to write a tcf fanfic but now I feel confused: is Cale able to breath underwater? Does he have a resistance against fire and burning? I definitely remember that during the fight with water spear in grey lake he was able to move and breath underwater for a long time but is it now one of his new skills? He also felt pain and regeneration from helping to stop the fire in jungles but logically isn't he supposed to be resistant to fire/light or purification? I am quite invested in this topic because it opens a whole new pandora box about how Cale is absolutely not a normal human being he strives to be LOL, this unlucky guy, man
No, Cale definitely doesn't possess the ability to breathe under water, nor any other sort of elemental immunity.
It wasn't exactly clear in the fight in the lake, but the way I see it, he either 1) lasted long enough through the fight not to need breathing since it was a very fast-paced confrontation, 2) had some sort of spell supporting him courtesy of Eruhaben or 3) surrounded himself by air with his wind power. The power of fire never protected Cheapskate from his own power, he in fact burned himself to death using it. Same with water, the Sky Eating Water has the power to manipulate water but not to the extent of being unable to be harmed by it. Just because you control an element doesn't make you immune to it by default, as is the case for most fantasy series (think pokemon for example). Same way waving a sword doesn't protect you against blades; unless you use your own to deflect them.
Also on the Cale being-burnt-by-fire scene, that is a good point BUT you have to remember Cale only had like, 3 Ancient Powers at that point? He had the Indestructible Shield, the Vitality of the Heart and the Sound of Wind, but his water power was in a bottle (Fire-Suppresing Water)... or did he already have the Dominating Aura as well? So 4? Yeah I think it was 4 in that scene in the jungle. It was still way before he got Cheapskate and Super Rock.
Anyway, the only powers that grant Cale any sort of protection are the Shield (obviously) and Super Rock (that one time when Cale almost exploded, and also the reason why the OG Super Rock was able to go into that monster tunnel connecting two continents). The second one is rarely utlized in the story, since Cale isn't a front-line fighter as much as support and AoE damage dealer so he doesn't have much use for it. The Shield protects him better, and has a larger range letting him protect allies, too. So it would be kind of redundant. I suppose you could also count Dominating Aura as a "protection" of sorts, since it can defend against god-level intimidation, but the point stands.
Cale might indeed be "absolutely not the normal human being that he strives to be" as you so nicely put it, but he is still made of flesh and as vulnerable to injury as any other person; the way Instant harms him proves it. It's well established in the story that just because he can heal doesn't mean he can't be hurt or even die from his own powers or their side-effects. Luckily, Cale has a ton of people willing to protect him, so usually the only person who can harm him is the one he literally cannot avoid: himself!
I hope this helps you a bit. Good luck with your fanfic! 💖