1. I can own up to my mistakes and give and take without a score card.
This represents two thoughts: That you admit when you're wrong (own up to my mistakes) AND that you can help someone and accept their help (or compromise when sometimes it's not in your favor and sometimes it is - the "give and take") without need to keep track of who "owes" whom the next favor. That means you wouldn't say "The last two times, I did what you want, and now it's your turn" or "hey, the last five times, I gave in to you three of them and it's my turn."
The problem with the sentence is that there's no visual separation between mistakes and give and take. Usually you don't use a comma to separate something like "I can sing and dance" (not "I can sing, and dance") but in this case the comma would add clarity.
2. I can own up to my mistakes without a score card.
This doesn't make sense. The score card refers to the keeping track of who owes the favor in the give and take, not the number of mistakes.