About trade in

It doesn't matter when the trade in kit arrives. You have 14 days to return your trade in to Apple AFTER you've received your new phone.

You never have to send your trade in until you've received your new phone and have had sufficient to set it up and prepare the trade in for return to Apple.

As long as the screen isn't cracked, depending on what you say is small, the trade value can be adjusted down. Only Apple can tell you once the trade has been evaluated.

If you have an Apple Store near you, I'd recommend doing the trade in store after you get your new iPhone. This way, you'll have no surprises and you don't actually have to mail the trade in, even if you receive a trade in kit.