Briefly: As I said above, you can see in the second chapter of Vedanta sutra, that there are arguments against other schools, including Samkhya and Yoga. This suggests that there are contradictions between some of the core tenets of each school, even if, on the whole, concepts from each school may often be imported to support the other. So, it suggests that a teacher may endorse the ultimate teachings of one school over another. And a great teacher like Sri Krishnamacarya may take portions from each and explain to his students what elements may be rejected if they contradict each other. The Six schools often disagree with each other in matters of fundamental metaphysical categories, the nature of the pramanas, etc.

Whether the Vyasa who commented on the Yoga-sutra is the same as the one who is spoken of as writing the sacred texts is doubtful.

Regarding the dating of the texts, I am not an expert, so I don’t want to claim too much. But generally, the Gita is held, by scholars and in the traditional sources to be dated earlier. The refrence in the Gita may not be to the Vedanta sutra as we have it (there were other sutras in currency than Badarayana’s). And the Vedanta-sutra may refer to Samkhya-yoga traditions without referring to the yoga-sutras as we have them either. The Yoga-sutras and traditional commentaries admit that there was an earlier tradition anyway, going back to Hiranyagarbha.

Take care.

The Gita 6th chapter was written before the yoga-sutra, so it is not Patanjali’s yoga, though it it seems that some elements of it were taken up by Patanjali.