Is there a gap between weak learning and PAC-learning?
I think you're confusing two notions of hardness: computational and statistical. 

In Q1, you point out, essentially, that boosting produces an improper learner. I think what you're asking then, is: Is there a concept class which is (computationally) hard to learn properly but can be efficiently learned via boosting? I believe there should be such examples, but can't think of one off-hand.

In Q2, please note that SQDim refers to hardness in a statistical sense -- i.e., the minimal number of statistical queries required. (We don't have any unconditional computational hardness results but plenty of statistical ones.)