See, I see that as badly written Donna, not so much Jason. My reading of those scenes goes like this:
Jason tries to stop a kidnapping. He thinks he's a good guy (who's willing to do what other good guys, read Batman, aren't), so stopping a crime makes sense. He doesn't try to kill Duela Dent, either because she was a Titan at some point and he still feels loyalty to that team (given he was upset when he found out they had no memorial of him, it's not entirely out of the question), either because what she does/did doesn't cross into what Jason deems killable offence. After all, she's not harming a kid in the process, so there's some basis for that as well.
He runs after her, presumably to arrest her, and he finds her corpse, and the Monitor who killed her and then babbles that neither she nor Jason should be alive. I can see Jason feeling that the Monitor impinged on his territory, and also that the Monitor, by killing someone that didn't deserve death by Jason's standards, is now a criminal Jason must stop. I can also see that the Monitor's words would make him wary, and he'd decide to do some research on the issue. Jason's not afraid to interact with heroes when there's knowledge that could benefit the hero community as a whole at stake.