First of all, only Hail and Sandstorm do damage.
Hail
All Pokémon except Ice types and those with the Ice Body or Snow Cloak abilities lose 1⁄16 of their max HP at the end of each turn.
Sandstorm
All Pokemon except Rock, Ground and Steel types and those with the Sand Veil, Sand Rush or Sand Force abilities, lose 1⁄16 of their max HP at the end of each turn.
So it's a fixed amount of damage, 1/16 of the max HP, which means it's neither Physical nor Special