How many resubmits of my thesis support paper can I expect from my sponsor and readers?
There’s no fixed answer here. Some advisors are willing to look at drafts in progress and pitch in a lot, while others only want to see almost finished drafts. Committee members beyond the advisor would probably only want to see one or two drafts to edit. Further, their comments will be more high-level than in depth. 

But the best way to make sure is to ask!