What to do when somebody gets removed from a group, but doesn't sync his account?

Nope, you made a ticket that's all you can do. they'll take care of it

Steam gifts will force synce his account after a couple of days, I guess you'll just have to wait till then.

First of all, I think checking people like him before inviting is a good option, secondly, if he got kicked, ask an admin to make a "force-sync" on him, next time he will login he will have to sync, and finally, if he win a give away, ask a new winner as the user is NOT in the group anymore.

By the way, is your group recruiting?

He will get auto-synced sooner or later, unless his profile is private. In the meantime, let your group know he's out, and that if he wins any giveaways to send a support ticket for a reroll.