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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.
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.
Heh, no problem. 
Yeah, the original broadcast in Japan was supposed to be 26 episodes, but the creative staff was informed that it was being cut down to 13, so they had to stop after one season. The Wiki article doesn't say why, but if I had to guess, it's probably because the show wasn't pulling in high enough ratings for the network it was on; Big O is one of those anime series (like Cowboy Bebop) which is much more popular in the West than it is in Japan. CN then co-produced another set of 13 episodes, which is the second season that's still airing on [as] today. That season apparently didn't perform as well as CN had hoped, so they didn't exercise their option for those additional 26 episodes. What that means to the plot is something I'm not really sure of...I don't think Chiaki Konaka has ever said if episode 26 was what was planned all along, or if it was changed when they knew they wouldn't be getting funding for more episodes. While there may have been a few tweaks, I feel like that was probably what was intended all along, since season 2 aired in Japan before the US, and so the creative staff wouldn't have known how things would turn out until several months after they were finished.
And yes, in the theoretical sense, the fact that the option for more episodes existed at one point means that someone could go back and fund additional episodes of the series. But at this point, given that it's eight years after the second season aired, and since CN has stated on a few occasions that they have no plans to produce more episodes, it's highly unlikely.
Yeah, the original broadcast in Japan was supposed to be 26 episodes, but the creative staff was informed that it was being cut down to 13, so they had to stop after one season. The Wiki article doesn't say why, but if I had to guess, it's probably because the show wasn't pulling in high enough ratings for the network it was on; Big O is one of those anime series (like Cowboy Bebop) which is much more popular in the West than it is in Japan. CN then co-produced another set of 13 episodes, which is the second season that's still airing on [as] today. That season apparently didn't perform as well as CN had hoped, so they didn't exercise their option for those additional 26 episodes. What that means to the plot is something I'm not really sure of...I don't think Chiaki Konaka has ever said if episode 26 was what was planned all along, or if it was changed when they knew they wouldn't be getting funding for more episodes. While there may have been a few tweaks, I feel like that was probably what was intended all along, since season 2 aired in Japan before the US, and so the creative staff wouldn't have known how things would turn out until several months after they were finished.
And yes, in the theoretical sense, the fact that the option for more episodes existed at one point means that someone could go back and fund additional episodes of the series. But at this point, given that it's eight years after the second season aired, and since CN has stated on a few occasions that they have no plans to produce more episodes, it's highly unlikely.
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