Can I complete the game just by doing the main story quests? Or do I have to do all of the side quests as well, to make sure my Hunter level is high enough for the story quests?
Meow-Oui. Yes to the latter question, but you can skip the ones you don't like on your later playthroughs (D: Oh Air Robo GP, how I loathe thee).
All quests are mandatory, but the order you complete them in is not. The WiFi/DLC quests (already preloaded on the NA versions) are optional. There should be ~80 quests or so.
It's suppose to give you the feel for the atmosphere of the game (such as the fun dialogues).
Don't miss out on the Tail Concerto cameos!