How many concrete piers do I need to support a 10 x 10 wooden floor supporting a 1600 lb shed?
As with any advice you get here, keep in mind that you must stay within the bounds of IRC building codes adopted by your local municipality.  Obtain a building permit and a local inspector will give you guidelines on minimum requirements for posts, beam spans, joist spans, etc...
Your concrete piers must be a set depth to go below the frost line in your area.  You must set the posts on top of concrete with a post base to protect from rot.
When in doubt always make it stronger!  Increase to 12&quot; piers with 6x6 posts.  Consider using 2x10 or 2x12 beams doubled or tripled on each row.  Make sure that each beam span is no longer than 7' if you are worried.  When you submit your plan for the permit then the inspector will point out any flaws or deficiencies that you need to address before you start.
NOTE: Any specific sizes I use are merely examples of how one can increase the strength of a structure.  Nobody can say for sure without more specifics about the shed design.

Do piers always need to be at the very end? Or can there be a small overhang of wood? Ideally I'd like to have the piers away from the edges of the perimeter so they're not as visible.

When the beam extends past the last post this is called a cantilever.  IRC building codes specify the maximum allowable length for live load for different types of lumber, and different sizes.