1) Experience comes from killing monsters and completing quests. It's possible to level up just by grinding monsters. That said... why would you? There are enough quests that you can level all the way just completing quests (since doing the quests almost always implies killing the bad guys along the way). The quests fall into a few broad categories, but the explanation texts are always well-thought out and typically interesting.
You can solo as much or as little of the game as you wish (with the exception of instances). Certain types of quests go faster with multiple people, sometimes that slows things down. Overall there are very few non-instance quests that can't be done by a competent duo; a few call for more, but these are uncommon. You can solo everything else.
In my experience, unless the quest called for killing a named and his bodyguards, most quests should be done solo or duo. Because you have to divvy up loot and exp in a larger group, you lose more than you gain in some circumstances. But a good duo is tremendous.
The exception I mentioned above are instances. Elite quests pretty much mandate that you put together a group of 5. Typically this is where the best loot and the best challenges lie. That said, you can hit level cap without doing a single instance. Though again, why would you?
3) Any legal combination is a good combo. I've heard in FFXI that you can hose yourself by choosing a "bad" race/class combination. It's physically impossible to do that in WoW, if only because race is relatively unimportant. If it's legal, you will do right by it.
4) I did quests almost exclusively, though I did a little grinding monsters (mostly while waiting for groupmates to show up). Some of my friends went very heavily into tradeskills, and did quite well from what they told me. (They claimed that, if Blizzard had been rewarding the top tradeskillers in the stress test, they would have had closed beta accounts afterwards. I believe them.) And a certain amount of the time you just spend exploring, for a quest or otherwise. I remember a friend and I were hunting for a moonwell to steal water from for a quest, and he stumbled upon a monument to Grom Hellscream, one of the Horde's greatest chieftains. You can find some really neat stuff in this game, so take a look around!