Way to see when someone added a game in their library?

Only the account owner can check that here: https://store.steampowered.com/account/licenses/ , others can't.

If the game already exists in their library then just re-roll. There's nothing to investigate because you have to activate all games and you're not supposed to enter giveaways for games you already own. He can't activate a game he already owns, so if he won your giveaway, then bought the same game later, he failed to activate your game, so re-roll. If he already owned the game then he shouldn't have entered the giveaway in the first place, so re-roll. In either case, the answer is re-roll. And you should immediately cancel the pending gift on Steam if it hasn't been redeemed.

In fact, the easier answer, now that I think about it, is to just cancel the pending gift and move on. You got the cv, he has the game in his inventory, so just take back your gift and it will have been him doing you a favor.

Surely it's in the SG database somewhere. If he's a user, he has to sync, and if SG keeps this data, maybe support can help you with that.

this is a tough one, if he owns the game and failed to redeem your copy and hasn't responded to any attempts to contact him I'd assume he doesn't want your copy and it is your copy again to do with as you will.

First thing, cancel the gift, since he can still take the game for his inventory, even if he now owns the game... Then make a reroll request - you can take a screenshot of your gift history, showing when you sent the gift. And you can show you still have the gift in your inventory.

Good luck! :)

Just ask to reroll (or better, delete it and recreate a new one since many entries might have the game or no longer be active by now), doesn't matter when he activated the copy in his library