I also plan to apply to Stanford ICME so I am interested to see what others have to say.
I think your profile is excellent already. What will probably make you stand out more are your letters or rec and your own personal statement.
Stanford Math program is completely different from what I have researched. ICME is definitely more applied math related. The core classes and structure of the program are very different. As an ICME student you can take math classes and work with the math department. It is very interdisciplinary.
As for transitioning from MS to PhD they have a little information on their website but not much. Masters students are not guaranteed to get into their Doctorate program, you have to reapply.