Ranks just equal how much influence someone has. This website explains it well.
Influence is mostly just for domain selection and sorting out the picking order. So naturally, higher up on the influence you are, sooner you pick. Going via the titles. So a high mayor, won't pick before a low count.
The best way to get influence is by spending money. $1 = 10 influence. But can get it other ways. Making CoE youtube videos/helping out on that wiki/reddit, making posts here. Get nothing close to spending money though.