Ok, so it’s gonna be a really close call if I can get to RE-UP 10. Do I have to click anything to redeem the reward, or will it automatically unlock.

It will update automatically when you reached Re-Up 10, finished campaign on hardocore co-op or higher, and got 1 milion points.

You don’t have to redeem anything. You will get the black phantom weapon skins and the gilded raam instantly in your inventory!

Good luck on getting Re-Up 10!

Are the black phantom weapon skins considered to be flaming and / or midnight omen ones?

No. Black phantom are those transparent black weapon skins. Flaming is animated fire. Midnight omen is orange / red omen logo.