The Black Dragon is above where the fisherman is (or was if you already bought his boat). IIRC there's a marker/star/twinkle at the entrance. If in doubt, click on shop markers. You can always back out and won't (normally) lose time for leaving a shop/business before interacting with anyone.i went there and rented a room but the key says moist maiden its right next to the shop
There's like a half-dozen quest-driven events that automatically trigger when entering/leaving a shop, but that's it and I don't think any of them relate to Daiyu/the Black Dragon. If you're worried about getting trapped into something you didn't want to do yet, Save before going inside and reload if you don't like what happens.