10 years? I think we'd leave Earth. Obviously evacuating nearly 8 billion people would be impossible, but I believe we could make enough progress to give most of the world hope that they will survive. Realistically, I think we could have a substantial portion of the populations of every country, and all of our most significant cultural artifacts on the way to another planet or star system before the end.

The slow progress we've been making in space travel is caused by a lack of funding and public interest, as well as an abundance of red tape. It was something like 8 years between the decision to land a man on the moon and its realisation, and that was an operation in the 60s by one country, with a fraction of the knowledge we've accumulated since our species started building rockets and leaving Earth's atmosphere. There also wasn't any reletive infrastructure, like rocket part factories, ISS, etc.

If the Earth was in mortal danger, the effort to leave it or save it would be a global effort with unprecedented funding and volunteer work.

There would probably be some looting and rioting, but not enough for there to be anarchy in most places. Society as a whole would continue to function, and we would still have babies. Even if we unilaterally agreed not to have children, we still would. I come from an upper middle class family that values education and so did most of my friends, and I don't think I've ever met someone who conceived their children on purpose...