Nope, only Carol goes Super-Saiyan, and only when overloaded with energy. Her powers are similar to the original Mar-Vells but not totally the same: he had know-it-all cosmic awareness, while she had a much better energy absorption ability.
The Kree are a very regimented and genetically engineered society that's stagnating. In the comics they've been the biggest power (1000s of star-systems under their control) in the galaxy for a long time (100k years) - probably since all the previous leading older civs took the godhood route (Asgardians, Olympians, etc). Currently their only real galactic rivals are the Skrull and Shi'ar empires.