In the books they camp out and the barkeep explains he’s working with Gandalf, gives them a note about strider, and I think Pippin gets petrified by the Nazgûl.

Can’t fully remember.

Oh also Frodo’s identity gets sussed out and he has to make up elaborate stories for the townsfolk.

The Nine likely knew that The Ring was being held by a halfling ("Shire, Baggins"). So they went to "nice, cozy, hobbit-sized rooms" the barkeep mentioned earlier.

That's my understanding. The ring only calls to the Nazgul if it's worn, so when Frodo fell he basically fired a flair. In the bar itself they no longer received signals, so they went to the rooms they had booked.

I'm also spitballing but If I remember correctly there likely was an Agent of Sauron in Bywater who saw or heard them. As to why they followed his tip instead of their instinct? Maybe plot armor ooor maybe the call of the ring is less like a ping emanating from one location and more like a smell that leads them in the general direction