That's reasonable but even in a world war, you need only put people on nuclear submarines to protect humanity from extinction. Martian colonies are not necessary.
I think at the point where you're talking about saving fewer than hundreds of thousands of people out of billions (<= 0.01%), it's functionally an extinction-level event for humanity.
Only in the most pointlessly absolute of interpretations.
Ten might as well be zero. 50 people (25 pairs) is necessary with complete control of breeding in order to have sustainable genetic diversity. Without complete control, hundreds are necessary in any given pocket of civilization.
Even if there were tens of millions of humans remaining, there is virtually zero chance of meaningful recovery since all of the low-hanging fruit of industrial-age energy resources have been mined.
Depends on your horizon, but if you look at how fast humanity progressed in 2,000 or even 500 years, I think a recovery from a million+ people in a few thousand years is very realistic, especially if knowledge is adequately recovered through books.
Only up until the technical level of the industrial revolution. Getting past that would require easy access to an easy to use energy source, i.e. fossil fuels. None of which are left. Today we dig kilometers under the seabed to access energy. A society just entering the industrial revolution will have not the knowledge, tools, or materials to do that.
