No, unfortunately Lymsa's Cove is usually just a resting point for travelers who wish to find a guide for the mountain, or to sail outwards. The darkened area I wrote about was the actual pier; it's closed because of the storm. You can check out the merchant with Ry or head to the inn with Madison! The storyline in Lymsa's Cove isn't available until you get some more stats, it's meant for when you come back again :)