So I'm running a lanny deck with kothh it's a clansmen deck that focuses on income manipulation to bog down opponents marshalling.

Question 1

Does kothh negate coin mint?

Coin mint location states: After you play coin mint from your hand place 4 gold tokens from the treasury on it then kneel it to move any number of those coins to your gold pool.

Kothh states: you cannot place cards during the setup phase other cards you control do not provide a gold bonus.

My thoughts: Coin mint would still work it does not provide a gold bonus it produces it, just like Tywin Lannister.

Question 2

Joffrey Baratheon states : If JB is only king in play he gains 1 ate for each gold on him. Response after opponent pays gold to Marshall a character put that gold on JB.

Does this gold go away at taxation phase? Or at the end of the round?

My thoughts: No. It doesn't mention that the gold has an expiration. Therefore it's cumulative.

Question 3

Tyrion lannister states: while TL is participating in a challenge all knelt clansmen you control are also participating in that challenge on the same side as TL.

Do the clansmen need the appropriate challenge icon to participate in the challenge? Ie if TL is doing an intrigue challenge then only the knelt clansmen with an intrigue icon participate with him.

My thoughts: No, all knelt clansmen are participating in whatever type challenge TL is doing. And all strength is applied as well as abilities.

Question 4

Does valar morghulis kill duplicates of unique characters?

My thoughts: no. It kills just one copy of that character. And if you have a character with a save another character ability it cannot be used since everyone dies simultaneously.

First off, congratulations, your thoughts are spot on for the most part.

1) You're right, Coin Mint is unaffected by KotHH. A "gold bonus" is only the yellow coin symbol with a (positive or negative) modifier in it.

2) The gold on Joffrey doesn't go in the Taxation phase, but note that you can't use it to spend for stuff either. It's best not to think of those counters as gold at all, they're really just generic tokens to keep track of Joff's current STR.

3) Correct, the Clansmen's challenge icons are not looked at.

4) Here's where you're slipping up. The thing is, between the initiation and the resolution of Valar's kill effect, there's the chance to save characters from being killed. You can use any and all appropriate save effects here, also characters that save other characters. You could for example use CS Ghost to save CS Jon Snow before Ghost himself bites the dust. Also, the question "Does valar morghulis kill duplicates of unique characters?" is misleading. No, Valar does not kill duplicates of Unique characters, but it kills the characters themself. You can dicard the duplicate to save the character it's attached to, though.
Here's a neat little trick, since you mentioned LotR Tyrion yourself: If you have Tyrion and Timmett on the board and Valar hits, kneel Timmett to save Tyrion. Then, Tyrion won't die (because he's been saved) and Timmett won't either, because he's now kneeling. Nifty, huh?