simple answer: both

The rating is based on experience, earning points per game. And based on skills, so training staff will bring you forward.

it is also important to note that it is not productive to upgrade your coaches piecemeal. every time you hire "coaches" you set all of their levels. so if you buy a coach with 5 offense 5 defense 1 tactical 1 kicking and 1 fitness. the next time you buy a coach those stats that are 5s already do not count as paid for and will cost you again in sigining bonus as well as wage calculation.

As Merc is counting sheep at the moment, I'll field this one.

The answer is no whenever you upgrade the coaches it's as if they were all set to 1, you just pay for the new coaches. Think of it this way, you were paying for all 1 coaches, then fired them and paid for all 5 coaches, and then 10, and so on.