Viewing MathJobs Applications as Living Documents? I am in the process of applying for tenure-track positions. I have been told that getting your file submitted early is useful, so I am eager to apply.

I have a final draft of my application materials, but I am potentially still getting feedback from mentors. Is it appropriate to start to submit applications now, with the potential of editing these documents mildly as time progresses and I get feedback? Also, most of my recommenders have not uploaded their letters, so I am curious if I should wait for a certain number to finish.
You can edit documents after submission, but there's no guarantee that the search committee will see your edits.  Some people might start reading them as soon as you submit, even if it's before the application deadline.  Or some others might download or print out the applications for later reading.  But in either case they might not check back later to see if there have been updates.  MathJobs doesn't automatically notify the committee when you make an update, and even if did you can't expect the committee to reread your application every time you make a change.

So it is best to wait until you are satisfied that the documents are in a "final" form.  If you do find a serious error or need to add something, you can, but it's possible that the committee will still work with the original erroneous / incomplete version.

You don't need to wait to submit your application until all letters are in.  Most search committees would probably just wait to read your application until the letters have been uploaded.  Or, if their process is to read the application separately from the letters, then they could start reading the application right away and then read the letters after they come in.