Have you been able to encourage group work in asynchronous classes?
In linear algebra last term, I used our learning management system (Blackboard) to put students into groups that I created, and assigned each group to complete a set of problems and type up their work. Since we were using Slack for class discussions/chat, I suggested that each group create their own Slack channel to collaborate on the work. I gave them deadlines for rough- and final-drafts, with some number of days in between each of those. [e.g. Assignment posted Sunday night, rough draft due by Friday, final draft due Sunday night.] Since I was also on Slack for questions, I was reading a lot of pre-rough drafts (which is more than I usually get in a face-to-face class).
How have you tried to present proofs online?
As a first online attempt, I made videos, stopping periodically to say "now pause your video to answer this part of the problem". When I do this again, I would like to try a feature of an online homework system that allows a video to automatically pause and a graded question to appear. [e.g. "For this problem, we're going to use mathematical induction. As a reminder, what are the steps for a proof using this method?" Problem opens up with various, jumbled steps for an induction proof, etc.]
What have you done for online exams which have to be taken in different time zones?
Last term, I made exams available for a long period of time (e.g. 8AM-8PM) to accommodate this and other issues.