Why does game list update in real time? Its exploitable

You can't enter the GA of a game you already have, but because your list is real-time update you could just remove it from the Steam Support tab to bypass and get the game back later. this is likely useless for public GA but may be farmable in exclusive GA group.

Removing the game from the Steam support page is not recommended by SG. (It will be recognized as an infraction.)
The recommended course of action is to "hide" the game from the library.

If the violation occurs in the first stage and the other party is the violator in the stage of handing over the prize to the winner of the SG, it is not necessary to provide the prize. (Subject to reroll)

All you have to do is simply scrutinize the winner in SGtools for violations.

You can also detect users who are doing what you consider "multiple wins".
It would then be possible to exclude such people from participating in the first place.

Steamgifts Tools - Activated Wins
https://www.sgtools.info/activation
Steamgifts Tools - Multiple Wins
https://www.sgtools.info/multiple-wins
Steamgifts Tools - Giveaways Manager
https://www.sgtools.info/giveaways/dashboard
Please refer to the SG add-ons and tools discussion for more information.

I could see someone removing a game from their account and then entering giveaways to win that game they already own. Then when they win that game, they can reactivate the game on their Steam account and trade or sell the key they won here, but I'm not sure what you mean when you say it could be "farmable in exclusive GA group". I guess you could cheat by creating fake giveaways, but it would be a hassle. You would need to pick a game that everyone entering already owns and have them all remove the game from their Steam accounts. Then the winner of the fake giveaway would reactivate the game on Steam and mark the giveaway as received.

I have a hard time believing enough people would work together and go through all the effort to make it work.

While it is technically exploitable it seems like too much work for it to become a widespread issue, and it's fix can introduce more complications than what it would theoretically solve. But hey, ultimately CG has the final word on any proposed change.

Regarding 1, I think SteamGifts already does this by just comparing with the users current library as well as their giveaway wins within SG. If you remove a game in your library, the giveaway win is still there on your account which SteamGifts assumes you have if you mark as received. At most you could do this once for a game you activated outside of SteamGifts.

Also I think the solution in 1 would cause people not to be able to enter a giveaway for a game if they tried it on a free weekend (not sure though).

Sure, this method works once per game you own, but it's a lot of work and requires that you also bought the game yourself independently. You're not going to make much money on the flip once you subtract out whatever you paid for the copy you're using for the exploit, and that's before you consider the value of the time you spent finding the giveaways, fiddling with your library, and reselling it. Bundled keys just aren't worth that much, so to make it worthwhile, you'd have to be winning a bunch of unbundled AAA games that you also happen to own, and also hope that the GA creator gives you a key instead of a steam gift copy.

I'm sure there's some user in some region with cheap games who made $20 or $30 like this, but if someone is willing to jump through that many hoops for a few bucks, I'm inclined to let them have the money.