Do I need to finish a mission to receive its challenge unlock

After completing more than 2 but less than 5 challenges, a second unlock will be revealed - while the first is a character perk, the second is a weapon. The weapon will be unlocked after 5 challenges have been completed. If you complete 5 challenges on your first run you will get both unlocks, otherwise you can replay the mission later to get the unlocks you've missed.

No...you can hit rewind story. It's R1 on PlayStation so I'm assuming it's RB on Xbox. I don't know about PC. It gives you the option to rewind story so that the events and outcomes are different. If you hit replay mission and select a mission, you can rewind the story back to there.