Is there something you're supposed to do with the chicken outside the hotel? Is there something you're supposed to do at the bar? Or are these just there so that Gabriel can talk about his previous adventures?

There's no game critical feature involving the chicken. It does have a function if you manage to activate the game's Easter Egg interface. Similarly, there's nothing vital at the bar although it is sometimes a good place to find characters of you're trying to gather complete records of each suspect's moped license plate. There's also another Easter Egg function available in there!