> 2. How would you approach proving whether it's currently impossible to make a version of BabyAGI that can succeed at the task mentioned?

What do you mean by "proving"? Proving anything on the behaviour of a non trivial neural net is not a small endeavour, and that's assuming you have access to the weights.

Are you expecting for someone to test all the currently known tricks on a BabyAGI-like setup to know if that's enough to achieve your task? That's likely enough to tell you that it's not doable with a given budget, but that's not a proof.