Most of the issues are already a problem on Mars, and the rest would affect earth and Mars at the same time. No less, the earthlings would be more likely to survive each of those situations because earth is way better in a general case.
You can argue the other way: If we figure out how to live on Mars, global thermonuclear war on Earth is not a big deal anymore, it becomes an option.
(at least in the Earth-Mars cold war of The Expanse universe)
It’s always been unclear to me how much a Martian civilization would improve humanity’s odds of surviving Seveneves, what with humans being involved and all. I think it’s the article of faith I’ve seen the least substantive critique of— maybe because it’s so depressing to think critically about.
BTW, on this note I can recommend The Ring of Charon[0] and its follow-up books. In this case, its Earth suddenly vanishing due to some ancient alien technology being activated by mistake. The book then follows the consequences for the advanced but still partially Earth dependent space habitats in the Solar system as well as people on Earth figuring out WTH their planet has ended up and why.
The open question I have is does a planetary colony help per se, once the technical problems have been solved on an emergency basis, and what’s left are social problems which are far too late to cure?