I can answer some of those, but I was a nub so my advice shouldn't be 100% trusted.
Questing vs. Grinding:
Questing is pretty big in this game, though like in almost all pure Questing games, there is a point where you do have to sit down and grind. Though, I do believe you'd be able to grind in the game, questing is easier to start out with.
Not sure if there are any, I never saw any, but then again, their Item Mall isn't open, so there might be pets in there...
First come first serve. Whoever lands the first hit gets the monster, after that it's "blocked" (think Trickster) off. Though there are cases where if you click at the same time you'll both fight it.
It's just shared, I think. You also get some money from being in a party, I think. Not so sure about this topic.
You have two bags, but they are really small, big turn-off. I suppose you could probably increase their size with the Item Mall though.
That's all I felt like answering. 8D