How important is it to submit papers early to a conference? Does the submission number in any way influence the way a paper is treated?  

For instance, is it likely that an early submission will be picked up by a member of the program committee, rather than being handed over to a subreferee?  When I'm going through a list, I do have a tendency to be more concentrated on the first few elements.

Thus, is it a good idea, knowing that I'll be on time for the conference deadline, to submit a good draft a few days/weeks before in order to rank higher in the submission number, then update with a polished version?
With a typical peer-reviewed conference, submission number has absolutely no effect on the way a paper is treated.  Lots of excellent well-established folks submit at the last minute, and paper assignment is generally not done until the deadline is passed.  Furthermore, paper assignment is often done with the assistance of a paper-handling system like EasyChair, which includes randomization and non-order-based heuristics.