So what happens when someone doesn't mark a giveaway a received but he actually received it?

From the http://www.steamgifts.com/about/faq

The winner has redeemed my gift, but not marked it received. How do I proceed?

If your gift has been activated or redeemed by the winner, and they have not left feedback after seven days, please create a support ticket to request received feedback. A support member will review and ticket, and if the gift is activated on the winner's Steam account, approve the ticket and update the feedback to received.

I guess the creator of the giveaway will make a ticket, give evidence that he sent the game, and support will suspend the person who didn't mark as received.

That being said, I don't think support ever ticked a "received" checkbox, I have no idea if they can.

Recently some asshole threw a tantrum and decided to mark all his won games as "not received" to show how badass he was, he was suspended but the games are still "not received".

edit : just read Bombaklats' post, turns out support can in fact change the status to "received".

Will the giveaway stay as Awaiting Feedback until he marks one of them?

Yes, at least up to 7 days. So far it is neither a non-activation, nor a feedback abuse.
User (I assume you are writing about the missing Detour winner) just registered 2 weeks ago and has a few games only. So either the site mechanics and/or the usage of a Steam key may be something new to him. You can try (or get someone to try) nagging him in Spanish.
Also, apparently he has not been online on Steam for 6 days, so maybe he can and will activate later. He still has some 4 days to do that according to the 7-day activation deadline (FAQ again, see all occurrences of seven).