What heroes have that perfect card where, if you have it in your starting hand, it comes down first every time?
For me:
Argent Adept: Alacritous Subdominant
Absolute Zero: Isothermic Transducer
Chrono Ranger: "By Any Means"
NightMist: Master of Magic
Unity: Swift Bot
****edit:
Chrono Ranger: Jim's Hat
Wraith: Impromptu Invention. There's never a time when there's no reason to play that.
Tachyon: Fleet of Foot
AZ: Onboard Module Installation
Ra: Summon Staff
Legacy: Inspiring Presence
K.N.Y.F.E.: Battle Experience
Mr Fixer: Dual Crowbars
The Naturalist: Natural-Born Vigor
Setback: Looking Up
The Wraith: Imprompteau Invention
Tempest: Localized Hurricane
Most of my picks are cards that get you extra cards, diversity is the key to victory!
Bunker: Ammo Drop
Chrono-Ranger: Displaced Armory/Jim's Hat
The Scholar: Bring What You need, or Don't Dismiss Anything
Fanatic: Absolution
Visionary: Twist The Ether
Haka: Mere, Punish the Weak, or Dominion
Fixer: really depends on who we're fighting. Just whatever I have on hand.
ExPatriette: Pride/Prejudice, Hairtrigger Reflexes, Speed Loading
NightMist: Amulet of the Elder Gods
Omnitron-X: Electro Deployment Unit, or a Plating.
Setback: Looking Up
The Sentinels: Unique Capabilities, THEN Team Communication
KNYFE: Focused-Conduit Blade
Parse: Snap Decision
Captain Cosmic: [REDACTED]
Sky-Scraper: [REDACTED]
Guise: [REDACTED]
I know there are matchups where you don't play End of Days first turn as Fanatic, but there aren't very many of them.
I'd say there are plenty. Any in which the heroes need to build up, or if there are things that happen LATER that are worse than what they start with (Voss with his airships, Akash'Bhuta, Omnitron, Gloomweaver, The Chairman, The Matriarch, Spite (don't want to kill victims!), Miss Information, The Vengeance Villains, The Dreamer, Kismet) and it's useless against The Ennead.
For a lot of villains I'd hold onto it for a time when it will do more good, or as a contingency play if things get real bad. For Iron Legacy first turn is probably a good time to play it though. 
It's definitely great for Dawn, Iron Legacy, and La Capitan first turn, that's for sure.
Not a big fan of using against La Capitan. 
I agree with:
Tachyon: Fleet of Foot
Ra: Summon Staff
Wraith: Impromptu Invention
There is no reason not to play these cards since they allow you to play another card.
Most of the others it depends on the Villain and what cards have been played.
Not a big fan of using it on Dawn, Citizens are easy to kill off as they have HP, and using it adds -1 Citizens to the trash, not always a great move in my experience.
Not if you start off with Truth, Anvil, and Tears...
The Sentinels: Team Communication, then Team Communication, then Team Communication, then Unique Capabilities, then Unique Capabilities, then Unique Capabilities, then whatever looks useful out of my 13-card hand.
I do Unique Capabilities first, because if you get any of the signatures in your hand you can't play them for free, so you don't want to draw 4 cards and risk getting one until you have all of them out or you don't have one in your hand.
If the Heroes want to build up, then you want to play it first turn so that you don't end up wiping your own board after you've built up.
Against many of those Villains I would play it first turn. The Vengeful 5 I would definitely play it first turn, because they start off with so much nasty stuff that would be a pain to leave on the table. Against Miss Information, you have a consequence free way of killing the first Distraction. Against The Dreamer you are skipping the whole first half of the game, which can be very dangerous indeed. It's definitely a first turn play against Gloomweaver, since you can't use it to kill his Relics, so you may as well use it to kill his Zombies.
Wraith has 1 HP, all other heroes are incapacitated, and Omnitron has Interpolation Beam in play.
There are very few times when you don't want to play it, but never say never =)
Pretty sure it says MAY draw on the card thus you can get an equipment from the deck out like say smoke bomb to rederect the damage from the lowest hero (you) to the highest hero (Still you) and reduce it by one thus denying the damage that would kill you. Of course if your spirit is not as warp as mine pray omnitron has very few hp left cause your pretty screwed at that point.
My main point for several of those is that it is better to save it for later (after Miss Information flips, when Gloomweaver gets a lot of cultists/pins/strength of the gave out, etc.)
Nope, Impromptu Invention says "Draw a card" as the very first thing on it. You would die.
Oh well damn memory serves me well for being too lazy to check up on the actual card