Will 2x10 joists span 10 feet between rims for a cabin?
How far you can span with a given joist size depends on the load, the spacing between joists, the type / grade of wood, etc. - but not on the size of the supporting posts.  There are lots of tables you can find online such as http://www.southernpine.com/span-tables/

I'd see what's done locally but I'd bet you're way better off with piers made of concrete poured into sonotubes rather than sinking wood posts below grade.  But if you do, you do need a footer under the post.