Okay, let me take a crack at these...
I can combine most of your questions into a single overall explanation. Both seasons of the series were created by the Japanese animation studio Sunrise; it was directed by Kazuyoshi Katayama and written by Chiaki J. Konaka, with the original concept and design by Keiichi Sato. The show was originally intended to have 26 episodes, but its initial broadcast was cut short to 13; you can read more of the details
here. (I actually didn't know some of this before right now.) The show's creators chose to end the first season on a cliffhanger, in the hope that they would be able to continue the series. Based on the first season's popularity during its US broadcast on Toonami, Cartoon Network decided to co-produce a second season of the series...in other words, they fronted some of the money for Sunrise to create more episodes. Because of Cartoon Network's involvement, Big O Season II represents the only true [as]-original series on the Action block, and it also means that [as] presumably retains the rights to air the second season indefinitely. The American anime distribution company Bandai Entertainment still holds the DVD rights for both seasons, but the fate of the TV rights for the first season isn't as clear: either [as] chose not to renew the rights when they last expired, or Bandai Entertainment themselves may have lost said rights, which would then revert back to the Japanese producers.
As far as the ending to the second season...well, there's the catch. Cartoon Network apparently originally had an option for 26 additional episodes of the series, but the ratings didn't live up to their expectations, so they chose not to exercise that option. The show's creative staff have never come out and explicitly stated what the ending meant, or where they would have taken the series afterwards. It's been up to those of us who have watched the series to draw our own conclusions, and it probably wouldn't surprise you to find out that there have been a lot of very lengthy theories written up. I've been guilty of one or two myself.