Do fire elementals attack with fire damage? I'm pretty sure they do. In fact I'd bet on it. I could even find out for certain to see if their damage is affected by armor. But going by the Hiegernomecon and tactical battles, it is not visually obvious to me. Please, just color-code the damage. Fire damage can be red. Cold damage, bright pink. Death damage, neon green. Look, I don't care what color scheme you use, just do it!
As for champions who can potentially deal multiple types of damage with one melee attack, just separate elemental and regular damage from one another. So one attack that does 20 cutting damage and 1 cold damage would appear as
in a tactical battle.