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Yes, check the new winner. Support hasn't checked the new winner - they were chosen at random from the remaining pool of entrants.

Always check any and all winners you have. ;-)

Always check every winner. I've even had a chain of rerolls (3 max so far) because the entrants hadn't been caught yet.

You never 'need' to check any winner. Whether you choose to do so is up to you, though the service is appreciated. As the above posters have mentioned, support approves the reroll itself, not the new winner, so it's just as random as the original winner.

Is rerolling for unactivated wins allowed? Because the relevant option says "Did not activate previous wins this month"
So they won't be rerolled if their unactivations are over a month old?

if you want to, sure.

Yes, always check your winners. Honestly I had to reroll like 3-4 times in a row once. Support just don't have time to check, its chosen at random.