Do triggers cause extra index updates?
It depends on if you use an INSTEAD OF trigger or an AFTER trigger.
It sounds like you want an INSTEAD OF trigger where you can prevent the default values from being INSERTed or UPDATEd and replace them with NULLs instead.
(An AFTER trigger would effectively cause the index to update twice.)
You can read more on how to implement an INSTEAD OF trigger here.