So there's two real issues here.
How to start near your Friends
The first is a question of start location. In WoW, this is wholly dependent on your race. As of Cataclysm, each race has it's own unique start location, which means that if you pick different races, you will not be 'born together'.
HOWEVER:
It is extremely easy to get to the capital cities, and you reach these at about level 5, after completing your starting zone. And once you're in one capital city, it is very easy to get to any other capital city.
So my advice: if you don't mind playing separately for the first five levels (should take about an hour or so, if you go slowly), play whatever race you want.
All roads lead to Orgrimmar, the Orc capital -- if your friends play Undead, there's a zeppelin in the undead capital to take you to Orgrimmar, and if your friends want to play a Blood Elf, there's transport from the blood elf capital to the undead capitol, and from there you can get to Orgrimmar the same way. Tauren also have a zeppelin (from their capital) to Orgirmmar as well, these days. Thanks, Powerlord
The Alliance has similar arrangements with Stormwind as the hub.
On the Tank / DPS / Healer trinity
First off, every class can do damage. While you might hear grumblings on the forums that one class or the other is 'better', Blizzard generally does a good job at keeping people even. Never let someone else convince you the class you picked is 'bad'. Blizzard does not make 'bad' classes. But everyone has a personal preference, and invariably these preferences clash, and so you get the "Nerf Wars" chorus, summed up by the pithy:
"Dear Blizzard--
Paper is fine. Please nerf Rock.
Sincerely, Scissors"
Damage Dealers
While all classes can deal damage, there are four classes that can exclusively deal damage. That is to say, they cannot tank or heal -- they simply don't have the options. These classes are the following:
- Mage
- Warlock
- Hunter
- Rogue
Healers
Likewise, there are 4 classes capable of healing, totaling 5 unique subsets of healer. They can cast heals on other players, resurrect dead friends, and otherwise mitigate death and damage. These classs are as following:
- Priest (Discipline)
- Priest (Holy)
- Paladin (Holy)
- Shaman (Restoration)
- Druid (Restoration)
Tanks
Finally, there are 4 classes capable of tanking. This means mitigating damage, taunting mobs into hitting you first (and the party later), and other 'defender-y' things.
- Warrior (Protection)
- Paladin (Protection)
- Druid (Feral Combat)
- Death Knight (Blood)*
The Death Knight is a special case, as they are not available to players just starting out. The class becomes available when one of your characters reaches level 55.
Class Conclusions
As you may have noticed, Paladins and Druids showed up on both the "Tank" and "Healer" lists. They are good candidates if you or your friends want to potentially fill any of the roles.
My personal suggestion is that selecting any of the 6 'hybrid' classes is a good choice for starting character, just in case you want to switch roles later. The other thing is that though these classes can spec for these roles starting at level 10, they generally won't be needed while leveling up via quests etc. Three DPS specs will likely progress much faster than a dedicated tank, DPSer, and healer. The exception is if you want to run a dungeon (called an instance) with your pals, where the three roles are strongly encouraged (more so at the highest levels).
However, starting at level 30, your character can purchase the ability to alternate between two separate talent configurations (often referred to as 'dual spec') which means you and your pals can stick as 3 DPSers while questing, and break into your roles when you go dungeon diving. Depending on the exact class, this may require at least carrying around a spare shield in your inventory.
In Conclusion (TLDR)
WoW is a very social game, and any three classes will work well together. Likewise, it is quite easy to meet up, so don't let starting in different areas prevent you from playing what you want.
It's your character. Play what you want. WoW will make it work.
There's little worse than sinking time and money into a character you're only half invested in.
Best Answer
Just the EU accounts.
Here you can find a Blue answer on the official forums explaining that achievements, pets and mounts can only be shared by same-region accounts.