Currently, Batman in the comics is still Bruce Wayne. However, there have been alternate timelines and universes where other characters have taken on the Batman mantle for brief periods or in specific storylines.
Not that I'm aware of. Comics come and go, and it's possible it's no longer in production or readily available. You could search on dedicated comic forums or websites for more accurate details.
The relationship is complex. Sometimes, Batman Noel urban comics might share common themes or villains with the mainstream ones, but they often explore the urban environment and its impact on Batman's adventures in a more distinct way.
Yes, Batman is often depicted as a playboy in the comics. He has a glamorous and wealthy lifestyle that includes attending parties and being associated with beautiful women.