Interpol doesn't have field agents.

She wasn't incompetent. She was figuring out things faster than the FBI (but they were hindered purposefully by Dylan). She even somehow linked the mythology of "The Eye" to all of it, when there really wasn't any reason to. The most unbelievable part of it though was that she would fall for Dylan when he was acting like the incompetent fool (and when he is not that good looking, no offense, Mark Ruffalo). Also the ending when she doesn't turn him in was not believable to me, as there really was not reason for her to fall for him enough to give up her professional and personal ethics. That came out of nowhere. She even guessed that Shrike was his father before he said so. Part of the point of the character seems that a female agent, especially from an outside agency, would likely be disrespected, and dismissed when she is smarter than the other people working on the case. I guess that bias extended towards your viewing of the character, funnily enough. The romance was bad, cliched writing, but I don't think they meant her to be incompetent.

I did mentioned cannot did I? Maybe my English is not good for you to understand.

Interpol has jurisdiction in almost every first world country, but Interpol cannot issue arrest warrants. Any arrests are made by national law enforcement in accordance with national laws. I am surprised that you worked at an Interpol office and thought that Interpol can conduct arrests anywhere

Interpol is just a liaison agency and agents cannot conduct arrest out of jurisdiction and even carry a gun. I worked in Interpol office before. Movie is fiction not reality.