Insecurity is part of being in a loving relationship.
Yep, she may be very sensitive to criticism. But It's your job as the BF to help her settle them down, IMO. She also doesn't seem to be able to stick with forgiveness. BIG Problem.
Your "jokes" about her driving and flash cards and such didn't offer much in the way of reassurance. . But her emotions are her responsibility. and she needs to deal with them on her own. Believe me, I know the kinds of harmless "to guys" ribbing you're talking about. Don't work with women.
It also sounds like she's doing the 18yo girl thing.

She's a mess. et it go.