How to work on a joint math paper effectively? Now, I am working on a joint paper with some collaborators. The research is about mathematics. 
This is my first experience of working in collaboration.

Another author has sent a manuscript about our work. The manuscript is about 40 pages. When I checked some part of the manuscript, it takes a lot of time to read the content and unfortunately, there are a lot of gaps and mathematical mistakes in it. 
In addition, there are also many typos buried inside the manuscript.

Although, I can fix some part and adding some details and proof, I still do not understand about 50% of the manuscript. 
I have to mention that we can communicate via email only.

The following are my questions:


How do you manage to work collaboratively in research?
How do you detect the mistake and fixing it quickly? 
When should we keep trying to understand and fixing the argument or rather asking the explanation directly to the collaborators? Is 4 days-checking without asking question too short?
How to judge``grey part", like simpler proof, simplified definition, notation, sentences, etc?  


Thank you for your advice.
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.