Armor can never be below zero. The enchantment Rust also doesnt fall off or deal acid damage if it would set armor below zero. Corrode markers can deal damage or fall off though.
to answer your questions:
Armor goes to zero
If the armor is destroyed, Rust stays on the target but will have no effect until they get more armor.
If they switch armor, Rust stays on.
There are some wierd interactions between rust and corrode markers though, like how revealing rust on a creature may do nothing but make it lose 2 corrode markers.
EDIT:
armor = max(0,ArmorBonuses - ArmorNegatives)
where ArmorNegatives include rust and corrode