If David had a reservation to go to Canada to cross international lines he would have to use his legal name to go through customs. Therefore when the travel agent calls to confirm his reservation she would request to speak to "Winston", which is his legal name and not "David" which is his alias.
Using his alias David Shaw to have an affair with much older woman, once Winston ran off with the woman's bearer bonds, the woman could have had the police dust for finger prints on items his touched, particularly the tennis racket to find out his real identity.
when Steven lied to Mohamed he tried to determine whether Emily's attacker was breathing after being stabbed in the neck by a meat thermometer, Mohamed should've known Steven was lying because anyone who's stabbed in the neck with any sharp object would be dead within minutes.
When detective Karaman shows Emily the pictures and the sketches of he murdered man, you can see the man stabbed on the left side of the neck on the picture (as really happened), but on the right side of the neck on both sketches.
When Steven and Emily arrive at Emily's family home on the morning after the attack, they are still wearing the same clothes from the night before, have clearly been up all night yet Steven is clean shaven.
On the subway, the conductor says "Greenpoint next stop. Transfer to the A, D and F upstairs." The Greenpoint Avenue Station in Brooklyn is on the G line and it has no transfer point to any of these three trains.
Since the 1990s and the elimination of the "Montrealer", Amtrak does not offer night service to Montreal, nor are there sleeper cars on the trains. The only overland train route to that city on Amtrak is the "Adirondack", a morning train that is all coach seats.
When Gwyneth opens the door to apt. 219 - sealed with yellow crime scene tape - she inserts the key into the deadbolt and turns it counterclockwise. Based on the door configuration this would have LOCKED the deadbolt but it opens smoothly...
When Michael Douglas leaves the apt to play poker with his friends, he was driving a Mercedez Benz. You can clearly see in the dashboard 7:30pm, while at the same time the killer enters the building. It doesn't looks like the killer is going to spend 2:30 hours waiting for the 10pm call.
The police dusted the house, elevator, access, etc because at that point they hadn't determined how the intruder entered the building. The police has to follow standard police procedure such as questioning Emily and Steven.
Steven tells David, David would receive fifteen years without parole if David were to inform the police on Steven's plan to have Emily killed. Even though only a judge could determine the sentencing, Steven was only intimidating him with a long prison sentence in order to convince him to agree to the murder plan.
While Karaman is questioning Steven in the kitchen, he points out that Steven's footprints are visible all around the body in the blood. We also have seen that when Emily left the kitchen she left a trail of bloody footprints. However, when Steven runs to the door to fix the lock, he miraculously leaves no bloody prints on the floor.
After the scene, where Steven opens a door for policemen, a camera pans down the spiral stairs. You can see two officers down there on the stairs obviously waiting for director's command and then start moving simultaneously.
Emily supposedly works for the UN, and, indeed, we see her in her office on one scene. However, towards the beginning of the movie, she is sitting directly behind the US Ambassador to the UN, and even passes the Ambassador information about the delegate of another country. UN staff would never do either of those things - they would only happen if Emily worked for the US State Department, not the UN.
Emily is wearing a white blouse and pearls the first day we see her. The next day, in the delegates arena, she is wearing the same outfit. However, when we see her on the train headed to David's at lunch time, her outfit and jewelry are different.