i looked through like 20 pages and used the search function but no luck. Since conspiracy cards are 0 to play can you use any conspiracy card regardless of faction?

Example: Cthulhu only deck with only cthulhu cards as resources. can i play a conspiracy card like this one?

I couldn't edit my post but it the same question applies to these cards also.

Unless there is a faction symbol (called Steadfast) next to the title of a 0 cost card (see example below), then you can indeed play any 0 cost card without matching the resource of the card.

In this example below, you must have 2 Miskatonic cards resourced somewhere in your domains to play Combing the Archive.

So snow graves, etc. are legal to play without paying any resources in any deck.

 Yes but it usually makes no sense to have them in your deck. Remarkable exception is Snow graves that helps against recursion with any fraction mix, beings of ib can also be fine, but im not that sure and havent played with them as Im not buying any dreamlands untill they're reprinted 60 or there are no other new cards (hope for the first as there are some nice cards there).

Also note characters that can enter play without resource match via card effect like:
The Terror of the tides
Dreamlands Fanatic
Master of the Bannhammer… i mean Myths (needs restriction?, rise cost to activate 2,3?, some kind of effect steadfast? this card is too stupid)
Black Dog
Intruder from beyond
Dirk Sharpe
Stalking Hounds
Rich Widow
etc
(hmm i wonder how strong no fraction neutral + best of those would be)

 While the cards you posted could be played out of any deck, they would be effectively blank since the effects almost all work on enhancing or requiring another card of that matching faction which won't be zero cost with no steadfast.