First, instead of looking at just your refinements (because his battle armor could be better), find out your actual initiative from the Character window. Second, when you shield your team, who shields first? That is the indicator of who has the higher initiative. If you do certainly have higher initiative AND shield first, but your mystery goes after his, then we can guess that it is due to lag/internet connection, or some other delay in which your Mystery was too late.

It should not, the things that affect initiative are refinements and battle armor rank from my knowledge. Did he increase his battle armor rank? Also, did you check and make sure that your refinements are certainly higher than his?

Yup, that would mean he has higher refinements, which is why he has more initiative.