1. I don't understand exactly the numbers on diplomacy screen, I don't know why there is a 0 between a rival and me who I have attacked while changing the view placing my rival into the center I see 9.80 between me and this rival.
The AI sees you at +10 to start mostly for envoy purposes. The AI isn't really capable of raising relations with you like you are with it, so it gets a head start and can start sending envoys earlier.
2. I don't know why I don't get 3 points after a mission succeeded as it was before this expansion.
Relations from missions are now capped. You can have a max of +3, or -1.5. Your current bonus is the outcome of the previous mission. So if you succeed on one, you get +3. If you fail the next, your relations are set to -1.5 (so you lose 4.5 relations). The reject button is now used for missions you don't want to do, and you suffer only a small but cumulative penalty for rejecting missions, unless you're at 0. So if you have 0 and you reject, your relations don't worsen. If you're at +3 and you reject, you take small hits each time you reject.
3. How does the game calculate diplomacy points? Can it be decreased or does it only stagnant or increase depending the relations? What are the exact rules here?
Game takes your total positive relations with all players and your total negative. If your positive is greater than the negative, points go up. Else, they go down. The bigger the difference, the faster the rate of increase.
4. What does mean the "AI relation bonus" in the last line on the tooltip about relation factors?
As far as I know it's the same thing as 1, the starting bonus the AI gets from a human player. That, or it's the bonus between the AIs.