The subtype Shapeshifter that appears on the type line is mostly there to reinforce the flavor. A creature card with changeling is just as much an Elf, a Dwarf, a Sliver, a Goat, a Coward, and a Zombie as it is a Shapeshifter.

You choose the target artifact or enchantment an opponent controls as you put the triggered ability on the stack. Once the ability starts resolving, you choose which creature card to exile from your graveyard, if any. At that point, no one can respond to your choice or take any actions until the ability is finished resolving.

Changeling is a characteristic-defining ability. It functions in all zones, not only while a card that has it is on the battlefield.

If an effect causes a creature with changeling to become a new creature type, it will be only that new creature type. It will still have changeling; the effect making it all creature types will simply be overwritten.

If an effect causes a creature with changeling to lose all abilities, it will remain all creature types, even though it will no longer have changeling. This is because changeling applies before the effect that removes it.