Vaishnava Culture: Cooperation among Varnas

                                                                  Q. 1. Also, if a man is born as sudra, is he not eligible to study the shastras, because the sudra's karma is to help all other varnas?

                                                                  As you stated in the initial premise of your question, one cannot be designated as sudra solely on the basis of birth, but by guna and karma.

                                                                  Q. 1. What do the scriptures including the Bhagavad-Gita and Srimad Bhagavatam have to say about "collective karma"? Our individual karma is generated by our own intentional and unintentional actions but what about the actions we perform as part of a group or as a society? Does responsibility get assigned to each one of us? Particularly, in the context of the planetary environment, we have collectively made impacts on the earth and are now seeing the consequences.