1. The offline mode has its own set of missions, and the online mode has its own set. There's a lot of overlap between the two sets, especially early on. There are a few online only monsters (Alatreon and Jhen Mohran I think) and one offline only monster (Ceadeus), but they don't make up a significant amount of content. Aside from those three online/offline exclusive monsters, you'll get to experience all the other monsters and maps in online that you would offline. Oh, online also has event quests with weapons and armor that you can only get by playing online. The only offline exclusive armor set is Ceadeus'. Oh, and in the online mode you get access to high rank materials to make high rank armor, which have different skills and a higher starting armor value than their low rank counterparts found in the offline mode.
2. You can dive straight into online, although one benefit to offline mode is that you gain access to the farm which can provide you with a source of income and items. There's no farm in online mode. Since you carry the same character between online and offline mode, it might not be a bad idea to progress in the offline mode to build up your farm at times where your girlfriend is busy or something.
3. It's a little challenging, but not too hard to learn especially if you're playing with her.
It's been a long time since I've played so I'm sure I made a few errors though, but I'm sure someone more experienced will correct me. Otherwise, I think it's a great game (one of my favorites on the Wii) and I'd recommend picking it up.