If you have a flying mount, you can fly right to it. It is in Crystalsong Forest. If you don't have coldweather flying yet, you can have a mage make a portal for ya to get there. There is also a quest to learn how to use the teleport stone in crystalsong to get there, but you have to be mid 70's to get that quest.
In Dalaran's Alliance or Horde Building
Dalaran is a magocratic city in the World of Warcraft universe. The city is used as a front to attack the Lich King's armies and is inhabited by humans, high elves, gnomes, and goblins.
To get to Dalaran, there are quest givers in the zone Dragonblight ( for the Alliance in Star's Rest, and for the horde, in Agmar's Hammer) that will teleport you to Dalaran. To use this way, I believe you must level 74 or higher. You can reach Dalaran by mage portal, or by warlock summons at any level.
You can either buy it in Dalaran, Outlands, or Storm Peaks! You can buy the mount in either Dalaran or outlands! 1000g for the 150% speed 5000g for the 280% speed
The current Dalaran is a floating city high above Crystalsong Forest in Northrend. It is the major city of the Wrath of the Lick King expansion pack, just like Shattrath was back in Burning Crusade days. It has all of the grand master proffessions trainers as well as vendors everywhere. It holds the Violet Hold, a dungeon.
Yes, Arthas Menethil is the Lich King. You can see a cinematic of the Lich Ling by clicking on the plinth of the fountain in Dalaran.
Lag in Dalaran is one of those things that has plagued WoW ever since it hit the live servers. There is not really anything you can do about it. It has to do with how many people are in the city at the same time. The servers just aren't quite capable of putting up with several hundred people in the same area at the same time. Wintersgrasp used to be the same way during battles, then blizzard put in mechanics to limit the number of people that can take part in the battle. This helped quite alot. One thing that you could try is upgrading your computer's RAM and Video Card. This is not a guaranteed fix though, as even people with top of the line computer systems can lag a bit in Dalaran.
You have to be lvl 40 at least (150 riding skill) but i only think you can get them in Dalaran. For 750 gold but if you do For The Alliance/Horde you can get a free one but it is black instead of brown.
Tier 9 armor is sold in Dalaran. It is associated with the Crusader's Colisseum. It is assigned to get a player ready to fight the Lich King at the Icecrown Citadel.
Emblems of Heroism used to drop from bosses in Heroic dungeons, Naxxramas, Eye of Eternity and the Sartharion encounter. They can no longer be acquired this way, however. The Emblem vendors in Dalaran can exchange an Emblem of one type for the type one lower (Frost -> Triumph, Triumph-> Valor, Valor -> Heroism) on a one-to-one basis. In the Dalaran Sewers there is a vendor that can immediately exchange Emblems of Triumph to Emblems of Heroism. This is the only way to get more Emblems of Heroism.
There is no "capital city" in northrend. There is however Dalaran, which is the major city for the continent. It consists of vendors, dailies, trainers, auction house, bank access, and seperate horde and alliance areas.
Open up the help menu by typing /gm. Then you will find an option for "character stuck". Click this, it will try to put you back on solid ground, or else transport you back to your hearthstone location.