It's clear that Todoroki left side is burnt but isn't it also his flame side. If he is able to tolerate the high temperature he himself generates from his own flames shouldn't he also have been able to withstand the burn when his mother throws hot water on his face?
I don't think Todoroki necessarily is impervious to heat. Midoriya notices when they fight that Todoroki's use of his cold power is limited to how much cold he himself can stand, and that if he would not refuse to use his heat powers he would be able to remove this limit. So if he's not immune to the cold he may not be immune to heat. However, his father does have the flaming facial hair which doesn't seem to bother him. I think it's most likely that he's immune to the flames and the ice that he creates but when he uses it too much or is affected by something in the environment he's affected normally.
Well he maybe wasn't trained t activate his ability yet andhe does still get cold and freezes even though he's part cold
It could have to do with the fact that he was still quite young and probably hadn't worked on his quirk for a long period of time and even if his body was able to withstand the heat from his own flames, the boiling water would of probably been extremely hot to of left a burn of that degree on anyone's face. But I don't think we'll ever really know why.