If Sony got the rights to a third party character like Big Daddy does that mean they own the character
No, they won't own any of the third party characters that appear in PSA. The rights pertain to SuperBot being able to use the character in question for PSA and that's about it. Even then, SuperBot does not have complete freedom to do whatever they want with said character. For example, if 2K Games, Irrational Games and/or Ken Levine did not like something about Big Daddy's representation in PSA, they could demand a change and SuperBot would have to abide by it. Ken Levine even went so far as to write Big Daddy's story in PSA.
on a similiar note do they have the rights for big daddy on a sequel or just the specific installment in the series?
That would depend on the terms of the contract signed by Sony and 2K Games. If I had to guess, it would be at most for a possible sequel. After that, Sony and 2K Games would have to either renew the contract or negotiate a new one.
No, they have rights to use the character. I was on some other forums and people thought that just because Capcom got rights to put marvel characters in a fighting game, Capcom owns Marvel
I'm not knowledgeable about video game business but even I know thats not the truth, in the same regard does nether realm own DC lol hahahahahaha. but on a serious note check out injustice it looks great