Alexstraza is a Dragon Aspect, similar to a god to Dragons, and rules the Red Dragonflight. She oversees life on Azeroth, appointed in her function by the Titans. In the game, she can be found at the top of Wyrmrest Temple.
That depends a bit from where you start to count. There's about three or four quests to do before you get an audience with the Dragon Queen Alexstraza, although I am not sure this is strictly necessary to unluck the Wrathgate questline. The Wrathgate questline itself is about five quests, where you need to go to Angrathar (the Wrathgate), and you will be asked to meet with Alexstraza again, and prevent the resurrection of the great dragon Galakrond. Upon returning, the battle for the Wrathgate starts. After that, Alexstraza in her dragon form asks you to meet with your faction leader. You will then receive a quest to go to the opposing faction leader and upon your return the quest to come along to Undercity. This is where the last quest takes place (The Battle for the Undercity). All in all, that is a minimum of about 12 quests, but if you want the entire background story as well (told in a few separate questlines) you'd be on 37-40 quests (all depending on whether you are Alliance or Horde).
The ratio for horde to alliance on Alexstaza is pretty even. Here is the latest census information: Total Characters: 23,366 Alliance: 11,665 (50%) Horde: 11,701 (50%) A to H Ratio: 1 : 1 A to H Activity Ratio: 2.1 : 1 (Meaning for every active horde character, there are two alliance characters.)