what is dragon-city's government? who rules them? i am confused in that area.

That's something I think you should ask the Magna Draconis About.

The greatest authority we've seen so far is DC Security. (There is the Dragon Council, but they give the impression of having very little public sanction to do what they do.) My guess is whatever leadership there is is either some expanded mayoral system, (Since DC appears to be very large and urbanized.) or a parliamentary something or other working out of Precinct or some high tower in Sun City.

We had a bit of discussion about this a long time back, though I can't remember where the thread is anymore. Anyway, they don't tell us much of anything about the government in the show. DC Security exists to enforce what few laws they seem to have, and in general, they seem pretty weak compared to police forces in the real world. The Academy has a lot of authority in racing matters, but so far it's not clear if they have any political power in the city. The Dragon Priesthood doesn't seem to do that much in modern times, though they may have been a force in the long ago past. Their economic system looks like it's based on capitalism which among other things means large corporations or wealthy individuals could have political influence.

The Down City Council appears to be an independent non-government entity that exists to keep the peace between the various Down City crews. It seems to be organized like a board council just to administer matters among it's own membership and nothing more.

So far I haven't caught any references to government officials (other than DC Security) or political offices, politicians, voting or a military. So I'd say this is all up for grabs.