Does "this attack" mean only when one of your characters is attacking? Or does it mean when a battle is occuring? IE, can you use something like Lost City to pump up a defender when it says "+3 +3 for this attack"

"this attack" refers to a specific attack interaction. you can use it at any point after characters are considered attackers or defenders.

CAUTION:

'this attack' refers to the specific attack occuring while you play the card - BUT some 'this attack' cards DO have the condition that you must be an attacker (eg: One-Two Punch) or a defender (eg; Nasty Surprise)

Lost City can be used for an attacker OR defender, as there is no such condition attached.

Thank you both.

Next Question. The case of Dr. Doom and a Doom Bot. Dr Doom attacks. Before the attack resolves, Doom Bot is used to ready Dr Doom. Dr Doom is then used with Mystical Paralysis to exhaust someone, thus exhausting Dr Doom again.

Does Dr Doom's attack resolve?

Next next question. Mr Fantastic, Stretch is on table. Random FF character #1 is stunned in battle while he has an equipment card equipped. Can Mr Fantastic, Stretch move the equipment card to another character before character #1 is stunned, and character #1 still benefit from the effect of the equipment card for that battle? IE, would overrun still be 3 less if Personal Force Field were removed from a character about to be stunned?

1) yes.

2) in battle, you are not able to play an effect during the resolution.

The only possibility is to use Stretch after the declaration but BEFORE the resolution of the attack (ie: at the same time Savage Beatdown and Nasty Surprise can be played), in which case the character #1 would no longer benefit from the equipment.