Yes public Baths and Onsens are still popular in Japan. If you ever go to Japan you should definitely try it out.
Regarding the episode, it is interesting that Hina still dies... she doesn't have anything to do with the gang, except that she's Takemichi's Girlfriend...
Oh and no, Takemichi doesn't have any memories of what happened when he goes back to the future. He relies on Naoto to get all the updates of what happened during the 12 years that he "skiped".