I'm running R&D on an IG deck with 12 face down cards in archives. I'm playing Edward Kim.

My run is successful and I access Hedge Fund which I trash. Do I have to pay 12 to trash it with Kim's abilities? Since operations do not have a trash cost, it can't be increased is my opinion. Also Kim's ability says "at no cost". Does "at no cost" overide IG?

Next case same IG deck, I play Drive By and find a Hostile Infrastructure. It has a trash of 5. And a net damage of one for trashing any corp card. Drive By also has "at no cost" for trashing an asset or upgrade,

Do I have to pay to trash the Hostile Infrastructure? The "at no cost" should at least take care of the normal 5 but would I have to pay the 12 for the IG effect? Since the "do 1 net damage" is an effect I assume it still takes effect.

Next case with the same IG deck. I access archives (and get hit with some stuff in there) If I play the card ( I forget the name) that allows me to pay to remove a card in archives from the game, Does IG's effect come into effect? Is removing from the game considered "trashing"?

My run is successful and I access Hedge Fund which I trash. Do I have to pay 12 to trash it with Kim's abilities? Since operations do not have a trash cost, it can't be increased is my opinion. Also Kim's ability says "at no cost". Does "at no cost" overide IG?

IG increased the trash cost, but Edward Kim ignores any trash costs so you trash it for free.

Next case same IG deck, I play Drive By and find a Hostile Infrastructure. It has a trash of 5. And a net damage of one for trashing any corp card. Drive By also has "at no cost" for trashing an asset or upgrade,

Do I have to pay to trash the Hostile Infrastructure? The "at no cost" should at least take care of the normal 5 but would I have to pay the 12 for the IG effect? Since the "do 1 net damage" is an effect I assume it still takes effect.

You pay zero cost. If there is a rezzed HI on the board you would take damage for any of those.

You would not take damaged from the HI you just trashed because it is not rezzed.

Next case with the same IG deck. I access archives (and get hit with some stuff in there) If I play the card ( I forget the name) that allows me to pay to remove a card in archives from the game, Does IG's effect come into effect? Is removing from the game considered "trashing"?

You can't trash anything in archives. Nothing happens here.

Thanks. Very helpful.