What to do when the 1 week limit is close for delivering GA, but I have a pending support ticket?

You reported the user or asked for a reroll? If you didn't try to ask a reroll.

Also don't bump your ticket, it only takes longer to answer it.

Always use the reroll request ticket in regards to any ticket where rerolling is a consideration- otherwise, even if staff answers your ticket in another category, they'll still require you to put in the reroll ticket before they actually respond to it (due to the reroll function being coded so as to require such a ticket be posted before it's able to be processed). Meanwhile, for things like suspending a user, any ticket'll work, so your reroll request'll work just fine for that.

In other words, don't clutter up the User Report queue with tickets you don't need to clutter it up with- save that ticket type for harassment, cheating, or similar issues, not for things directly related to your giveaways. Where a Reroll ticket isn't appropriate, and the ticket is not only in regards to the user's misbehaviors, use the Other category instead [note that only a small handful of staff can access user reports, while all of them can access Other tickets].

Other and Reroll will be your main ticket categories you'll use, by far.

As far as giving your game to a rulebreaker- unfortunately, that's a risk of public and group giveaways. As noted above, there's a chance you'll still get the reroll if they haven't been suspended for the infraction yet, or have been suspended for it within the past month, but otherwise.. no luck.
If you want to exclude rulebreakers altogether, you'll have to use a SGTools filter and post your giveaway on the forums (or inside a public giveaway, or given out manually, or so forth).

When you've gotten all the help you need from this thread, you can close it by clicking the button in the top right of the page. :)