In general, there are certain story quests in FFXIV that can be redone. However, not all of them are repeatable. It often depends on the nature and requirements of the quest. Some quests may only be available for a one-time completion to maintain the game's progression and narrative flow.
After returning to FFXIV, the story quest could be initiated by visiting your class/job trainer in the corresponding city or by checking the notice boards in major hubs. Sometimes, you might receive a message or prompt directing you to the right place.
It varies a lot depending on your playstyle and how much time you can dedicate. Some people finish it in a couple of weeks, while for others, it might take months.
Yes, in most cases you can still get the quest rewards even if you skip the story. The rewards are often tied to the completion of the quest itself rather than fully experiencing the story part.
The story of FFXIV 1.0 was a mixed bag. Some aspects were engaging, but it had its fair share of flaws, like inconsistent character development and a confusing overarching narrative.
The Heavensward expansion has an amazing story. It focuses on the Dragonsong War and the relationship between man and dragon. Estinien's story arc within this expansion is really engaging. He starts as a dragon - slayer with a one - track mind but as the story progresses, he grows and changes, learning about the deeper issues between the two races. And the landscapes in Heavensward add to the overall atmosphere of the story.