Yeah, ARR quests can all be completed as any class - obviously anything with a battle component will require you to play as a battle class at that point, but crafters and gatherers are free to do the final "talk to the person with the green tick icon" and collect the EXP reward.

And of course the majority of sidequests in ARR are unlocked based on you having a class at the right level to complete them!

Though if you're not up to Coerthas in the MSQ then there's no real reason to be there yet, except maybe DRG job quests. It's better to do sidequests in the same areas the MSQ sends you, instead of working ahead, since they're usually at least somewhat relevant to the plot - or happening alongside it, at least.