Question: Why did Mitch tell his firm about being contacted by the FBI in D.C.? What purpose does it serve him? It probably only attracted more attention to him.
Answer: He knew the Firm was watching him and would probably learn about the meeting. To head off suspicion that he may be cooperating with the FBI, he voluntarily tells his superiors about it, all while acting naive and showing he has nothing to hide. He may also be displaying a (false) willingness to be drawn into any nefarious Firm activities, allowing him to learn more about it.
Question: What happened at the end of the movie to the man who had been shot in the leg by the private detective?
Answer: When Mitch McDeere gets off the elevated train, when he's leaving Mud Island, a guy with a limp, wearing an overcoat (with faux fur), follows him. That's the guy. Mitch had never seen him before, so he gives him a weird look when he accidentally bumps into him. The guy with the limp was likely in the hospital while the security staff were doing their shakedown of Mitch.
Answer: He's not sure if he believes the FBI, and if they're telling the truth he doesn't want the Mafia discovering he'd been talking to law enforcement behind their back. This way he can avoid taking sides just yet.
Brian Katcher