Fire can burn ice while water can put out the fire.
Why is it called "Ice Berry" and why it can heal the burn? This doesn't make sense.
I think its cause you would put ice on a burn to help it cool down and heal irl
Yes, this is most likely the case. Keep in mind that a burnt Pokémon is a, well, burnt Pokémon, not a Pokémon that is on fire.
Also Game Freak Logic.