Does the Super Pet achievement require 1000 levels in addition to 1000 strength/health/defense/speed? What about intelligence?

I'm not entirely sure about intelligence, but I believe it does include level. I want to think it's just all of the ones that have achievements of their own in the training section (which would be Level, Health, Strength, Defense, and Speed).

i know for a fact that 1k intelligence doesn't count toward it!

Level does count. If your pet is that strong you kinda want your pet to be that smart anyways, but I agree with if it's not one of the Achievement stats, it is not required for this one.