You can find out what all the ranks and grades are, and what you get at each, by clicking on the bar at the very top that shows your experience. A window will popup that indicates your current rank and grade, as well as all the rest of them, and the rewards at each. You can also check the diplomacy ranks from a tab in the same window.

You don't have to be in space to rename your ship, just bring up the character display, and select your ship. Note that if you go to the Ship Selection officer and bring up the display to list all your ships, the same button will be used to "Dismiss" them. Just make sure you're not near the Ship Selection officer and bring it up.

There are 10 levels from each rank, which go from 1-10 in that rank. Level 11 in the rank is the requirement for level 1 in the next. You must go through the promotion ceremony to go from Level 11 back to Level 1 for the next rank, so make sure you spend all your skill points and then go see Admiral Quinn to get the promotion mission.

And honestly, every single time I forget and go back to try to get promoted at Level 10. I then remember that I have to wait for 11. Oh, well...