What happens if the Drowned God character is a unique already in play (due to effects like Head on Spike or Crone, etc)? Does he enter play as a dupe?

Yes, it would enter as a dupe, although the 'knelt, participating as an attacker' would fizzle. Then afterwards you may 'trigger' the dupe to save from the return to hand / bottom of deck, OR allow the character to go to said destination (and therefore discard the dupe anyway)

If you use Old Wyk to put a duplicate into play, the rest of the effect will fizzle. It will not attempt to bounce the duped character, because Old Wyk's ability didn't put that character into play. The character card that was in play, and the character card that you attempted to put into play with Old Wyk, are two completely different, unrelated cards. Old Wyk will not affect the character in play "by association", no. And it will not bounce the dupe, because the effect says "return that character...", and the duplicate is not a character.

Yes, it can still participate.

A character only needs the corresponding challenge icon when it is DECLARED as a participating attacker or defender. Old Wyk is not "declaring" the character; it is simply dropping it into the challenge as a participating attacker. Since it is not using the "declare" mechanism, it does not need to meet the "declare" requirements - only the requirements of the ability being triggered (which does mandate a power icon in this case).

How exactly is this part of the card ability "put the top Drowned God character in your dead pile into play" resolved?

1. is it checking the dead pile from the top to the bottom and first Drowned God character is picked? - or

2. it checks only the top card of the dead pile and if it is Drowned God, it resolves further, otherwise it fizzles?

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