1) I think you're referring to the way the races are arranged geographically. Either that or the way the servers are divided up, I can't tell. It is the case that there are 3 starting areas per faction, for two factions, so six "realms" altogether. If you're talking about servers, see Volondo's post. However, to be a little technical, you don't ONLY fight the environment on the PvE server... there IS consentual PvP, just not a lot of it.
2) There are more than 3 melee fighters! As Volondo mentioned, there are quite a few classes who rely heavily upon melee combat. The Warrior, Paladin, Shaman, Rogue, and (to a lesser extent) Druid spend a lot of time in melee. The Hunter, on the other hand, is a ranged damage specialist who avoids melee when he can. That's what his pets are for.
As to which is "best", Blizzard has done a pretty darn good job of balancing. It depends on what you want. Rogue has the highest damage output and plays a fast-and-furious style, but requires a bit more preparation and guile. Paladins are almost impossible to drag down, but take substantial time to defeat their enemies. They're not ONLY good against undead, they just have extra spells to use, but they can acquit themselves well versus anyone. Shaman and druids spend quite a bit of time in melee, but as hybrid classes they also rely upon direct damage, healing, and a variety of buffs and utility spells. The warrior is the character people seem to default to, for obvious reasons: he does good damage, can carry almost any weapon, and wears the heaviest of armors. The Rage system is unlike what the other classes use to power their abilities, but most people adapt to it well.
All the classes described above can handle monsters two levels above them if you play them with skill. With good equipment and a high level of skill you can defeat monsters three or even four levels above your own, and this does NOT depend on class. As I said, they're very well balanced.
3) Okay, let's be a bit more precise about our terminology. In WoW, your level affects a few categories of things: your spells/abilities, your talents, your stats, and your tradeskills (to a lesser extent).
When you level up, you get access to new spells and abilities. You must visit a trainer and pay money to get them, but then they're yours at full functionality.
You earn talent points every level starting at level 10. These are independent of your normal spells/abilities. You can spend these points to boost existing abilities or to acquire new, special ones. There is no cost to use your talent points, and the effect is immediate.
Certain stats (Stamina, Intelligence, etc.) go up with your level, and certain stats (primarily weapon skills) have their cap raised as your level increases. For example, if I have the swords skill at 95/95 and level up, my maximum swords skill becomes 100, so I'd be at 95/100. Increasing these skills is simplicity itself-- fight a few monsters and swing your sword at them and your skill increases on its own.
Tradeskills can normally only be increased by using them, i.e. if you're a blacksmith you can only increase your skill by making swords and such. However, your ranking in your profession can only peak out at a certain point, say 75. Once you get there, you have to go to a trainer and buy the next "level" of tradeskilling (say, Apprentice Blacksmithing up to Journeyman Blacksmithing). This requires money, and it also has a level requirement to prevent certain types of twinking.