How to work on a joint math paper effectively?
As mentioned before, by other answerers, there is no way to answer this question accurately - it is a matter of taste, and the preference of  who you are working with. The best suggestion I think I can give is to use Overleaf

You need to work collaboratively on maths - Overleaf uses LaTex as its back-end, and allows real-time collaboration. Google docs also allows realtime collaboration, but you will likely suffer from formatting issues. Change control is done with git, which means you can also work offline and use git for conflict resolution in the content. You can use either the gui or the command line.

The rest of the questions (2-4) are purely issues of this particular paper, and thus impossible to answer definitively.