[Suggestion] More possible options or feedback for Not Received

The listed game is either recieved or not recieved, that's all there is to it. The only other option that I'd consider would be a pending/hold option for pre-orders that aren't yet available.

If winner could mark “Broke key” then I could note the status change and update key asap.
If winner could state “Owned” then I could directly raise a reroll ticket.

Winners can already do this in the comments, there's no real benefit to programming an alternative system for it.

If anything, there should be a notification for the creator when one of his winners marks a giveaway as not received, so the creator can evaluate and resolve the situation, including reaching support if appropriate.

I don't find anything positive in extending not received to another degree, it doesn't really matter for anybody, the option is here to prevent abuse for both the creators and the winners, not to provide feedback about the key being redeemed. By definition you shouldn't create giveaways for stuff that isn't redeemable in the first place, and in the perfect world there should be no "not received" at all. SG should not encourage users to post giveaways they're not sure about by highlighting that not received is fine when picked with "the key was region locked" or "the key was already used".

I was thinking that in the giveaway winners, there should be a private posting section to make communication easier.

The thing is that in the end, the game wasn't received.

When you get an invalid key, do you receive the game? No.
When you get a region-locked key that doesn't work in the region you're giving the key to, do you get the game? No.

There's a reason there's only "Not received" and that's because it shows a precise outcome. That the winner didn't receive the game. I personally don't think we need to muddy the waters by adding these "Sure, I broke my promise, but it was because the key wasn't what the winner needed, pls be lenient" options. The outcome's always the same and in the end the reason's also always the same as well.

I sent a key but winner already owned the game

This one should not be marked as not received, this one should be rerolled.

As the "not received" feedback can only be selected after 7 days have past, is this really a common occurrence for winners to wait 7 days then mark "not received", instead of commenting immediately about the problem?
I've had winners sitting on their key for over a week while still appearing online all the time (yay...), but I've never had someone who waited until the "not received" feedback becomes available instead of immediately commenting about a key issue. That seems... quite irrational. And I'm not sure people who do that would be that proficient at picking the proper "not received" reason, would that feedback be made more detailed.

Winner ALREADY have a feedback option - giveaway comments. What really CAN be useful - is a way to message the winner directly on site, and even that's not necessary, because there is a email for that, and you can always add winner on steam.
And, btw, "I sent a key but winner already owned the game" is not a valid case. Good practice is to check if winner has the game on account before giving them the key (It can be an issue with giveaways of >50 copies though).

Many commentors are writing that its pointless to have extra options for Not Received, yet they don't address the possibility that the winner may not communicate the reason for "Not Received".

So let's actually explore what can be done or can happen in that case:
If the winner marks GA as "Not Received" but does not leave a comment on the GA regarding the reason, then the only other option is for the creator to either leave a comment on the GA in hope the winner will notice (unlikely) or email the winner directly with the user's email address on the GA's winner page.
The winner would respond to the email with the reason for Not Received, not reply at all, or the email is invalid & bounces back. The former would allow possible resolution.

Isn't it safe to say that the vast majority of users entering GAs are often motivated by receiving free games to at least leave a comment or respond to an email they receive? Or does this happen more often than people assume? Is non-response more common with certain types of GAs: Public, low level, low points?

On a different note...

update key asap

To be clear, this can only be done from the GA winner page, but can the key be updated before clicking Sent? I forget.

Keys that are mistaken or invalid should be marked "not received." It is up to the creator as to how he or she wishes to handle things after that.

Keys that are owned should be re-rolled. They should not be marked "not received" because the key actually was received. The proper response from the winner is to leave the giveaway unmarked and notify the creator. It is the creator's responsibility to check ownership of a game before giving the key to the winner.

I can think of only one situation where additional feedback may be warranted, and that is when the creator gives the wrong version of the correct game. (Remember Pirates of Black Cove Gold?) Fixing that requires communication between both parties, however. Ticking a box isn't going to be enough, so there's no need for another box.

More importantly, any creator who gives keys to winners without initiating contact is opening him or herself up to trouble. Why would you not contact your winner before handing over your key? Wouldn't it be better to check if he or she is still alive and responsive before turning over the key? Don't you check the status of your winners before handing over the game? Surely, you want to at least check that he or she owns the game, first. You don't have to be a moderator to get someone's attention. Just hold onto the key for a day or two. In most cases, the winner will contact you asking about his or her win. If not, you can send an email, wait for a response, and if there is none, ask Support for a re-roll (due to non-response from winner).

Either they got the game or they didn't is plenty of options. There are avenues of communication and if that fails, support can help.
No need to complicate things any further.

I don't think it's necessary, either. If someone receives a key that doesn't work or even already owns a game (which in itself is odd, except for DLC, everybody makes mistakes ;>) and does not bother to get in contact with you to resolve that issue, I highly doubt this person cares to distinguish between why he marked your GA as not received anyway. Might come across like a hassle, but just checking the winner's SG or Steam profile for activity and maybe their previous wins via SGTools can save you a lot of trouble, at least that's my experience. Anything further should always be handled manually IMO, since these things tend to get complicated very quickly, and I have to commend SG staff regarding solving such problems, the few cases I had were always easily and swiftly solved.

Let's needlessly complicate a simple system.

Edit: A simple system that already confuses a lot of users

would be nice to get a notification when a game you gave away is marked as not received, insted of manually checking pages and spot one with a red X...