If you kill ships on your own side you lose score. AFAIK that's the only way to end up with a negative score.
Rearming and repairing reduces the score 20 points. I tested it by doing the first mission without any kill, and I finally got -20 points on my score

But I have a question : when I kill an ally fighter, I lose points, but even I jump out, I don't succeed the mission because of treason (and so, I can't go to the next mission). If I restart the mission, is my score getting back into its last value ?
Another question is about on of the mission of the first loop of the FS2 main campaign : Love the treason...
In this mission, there are something about 15 Hercs Mark II, and some of them are destroyed, but it still be difficult in Insane mode engaging all of the hostiles that didn't die. So, before the Hercs are getting hostiles, I kill 2 of them (3 is out of time), and then it's easier to destroy the other Hercs. Do I lose points when I do that ?
The points (sub)system of Freespace has been very well thought
Not at all. There's no high-score board for me to type three-letter swears in. 
Yeah, but the score is for getting higher rank ^^
I have another question about those ranks : if I'm Admiral (9,000,000 points = 900 Sathanas destroyed with 100% damage

), what are my privileges ?