The thumb drive thing was a dark obvious pun that didn't require deduction.
How did he deduce the murder weapon? He didn't even know what it was until the cop told him. It was a bloody object in the home of the serial killer of course that was the murder weapon.
What do you mean uncovered Annika? Do you mean found her dead body, because Catwoman did that while robbing the drug deal. She also interrogated the dirty cop involved in killing her to get a recording of the rat basically confessing to being the rat. Batman and Gordon weren't even considering that guy at the time.
Edit: He did find Annika alive by following Catwoman from the club. Then instead of interrogating or protecting her, the witness that was with the Mayor on camera, he follows Catwoman to wrongly accuse her of killing the Mayor.
He didn't put the conspiracy together, Riddler literally exposed the corruption. That was his whole plan, to expose everyone connected to that case that was working for the rat. That included the police commissioner. Of course the police were involved. He literally saw a cop that he beat up at the club in the police station. It doesn't take a detective to see crooked cops working for gangsters and the police commissioner get killed and think maybe they are involved in something shady.
Batman didnt actually care about being a detective initially. Man just thought he could go out there and beat people up and make Gotham a better place through fear. He was just going out there and being a mentally unstable emo dude for 2 years. No strategy. Its just now through this(his first tough serially liked BIG case with actual long term consequences) he has realized he has to be a much better detective to see things from a bigger perspective. So yes. He was a goofball before this movie. This is what is gonna put him on the road to becoming the Batman we expect. You gotta suck before you get better.