You pay all costs and follow all timing rules for cards played this way. For example, if the exiled card is a land card, you may play it only during your main phase while the stack is empty.

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 a Kithkin, a Goblin, a Merfolk, and a Brushwagg as it is a Shapeshifter.

If an effect causes a creature with changeling to become a new creature type, it will be only that new creature type (unless the effect says "in addition" or similar). 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.

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