Do cards with a cost of 0 require you to exhaust a Domain to play them?

I would have thought not, and indeed wouldn't even have asked the question, if not for the Sword Of Y'ha-tallo.  Which, as I'm sure you're aware, has a printed cost of 0 but two Cthulhu faction symbols in front of it meaning, if I understand the rules correctly, that you must exhaust a Domain with at least two Cthulhu faction cards on it in order to bring the sword in to play despite its 0 cost ... ?

No, you do not need to pay for a 0-cost card, so you do not need to drain a Domain.

What the faction symbols next to the '0' indicate is that the card is Steadfast.  To play a Steadfast card, you need to have that many resource cards of the represented faction under your domains.  You don't need to drain them.  This mechanic effectively prevents you from incorporating some random faction's 0-cost cards into your deck without actually playing that faction (as opposed to the good ole' Servant from Out of Time from the CCG days which was a 0-cost Yog card that found its way into pure Agency/Misk decks, etc because it wasn't Steadfast so could be played without needing any Yog resources at all).

Ah!  I get it now!  Thank you ...