While @mafutrct handled most of the essentials, I figured I would address some unanswered portions of your question.
Why do we start backwards? Well, this one is a little bit tough, and has more to do with the Japanese culture of ranks more than anything; but if I were to venture an educated guess, 級 (kyu) ranks count down because they are measurements of strength: an ordering if you will. Consider the following example:
Players ordered by strength
- Person A
- Person B
- Person C
In this example, we can understand that Person A, listed first, is the strongest of the three. Kyu ranks are similar to an ordering like this. So...
Why do we start to count upwards again? Well, like @mafutcrt mentioned, once you pass the highest rank, and become "dan" you are more like a master (this is the exact same system used in Japanese martial arts -- dan ranks are usually translated as "Nth degree black belt" in English). The Japanese character for "dan" (段) also means "stair", so each increase in rank can be thought of as one more step up the staircase of mastership.
What are the basic differences between ranks used in different countries and different servers? See this worldwide comparison of ranks on Sensei's Library.
So how is this all related to rating? By rating, in contrast to rank, I assume you mean strength, such as is represented by scores in the Elo rating system, among others. The simple answer is that in many systems, such as most online Go servers, there is little (if any) difference, because you can go down in rank just as well as you can go up. In others, such as with ranks given by the Japanese Go Association, you can never go down. In these systems, rank and rating are somewhat disconnected, but there is still generally some level of correlation. Professional ranks are very different, and are discussed in the question Are professional ranks representative for how many stones of handicap players can give each other?