What happens when a Doppelganger is put into play from the discard pile?
Well, let's say that you are trying to use Unspeakable Resurrection to put Doppelganger into play. Since Doppelganger's ability is only when you play it from hand, putting it into play from the discard would not allow you to use its ability.
Type: Event Faction: Yog-Sothoth
Action: Choose a character in your discard pile with cost X or lower. Put that character into play.
So since "put into play" is not the same as "play", then anytime you put it into play, X would be treated as '0'. Doppelganger would essentially be a 0-cost, 0-skill, icon-less character with no relevant text.
From the FAQ:
(2.1) “X” (The Letter “X”)
Unless specified by a preceding card, card effect, or granted player choice, the letter “X” is always equal to zero. For example, Darrin controls Shadow Company (Summons of the Deep F89), which has an “X” for the card’s skill value. His opponent, Tommy, targets it with Called By Azathoth (Summons of the Deep F9). Since the Shadow Company has a blank text box, it does not have a value assigned to X. Thus, its skill is now zero.
When a character is insane it is considered to have no icons, skill, or text box, would Gibbering Soul's ability trigger if it's killed while insane?
We know that characters that are blanked don't do anything when entering the discard pile.
From the FAQ:
(1.12) Card States
If a card has an ability that triggers in response to said card entering the discard pile, that effect does not resolve or trigger if that card had blank text due to a card effect, was attached to a domain as a resource, or was insane before it entered the discard pile, unless that card specifically states otherwise.
In this case, the answer is similar. Since "destruction" would happen when the text box is still blank, it wouldn't have an ability at that point to be able to respond.