In the dinner table scene following the officers funeral. Frank and Henry are seen in their white dress shirts as part of their uniforms while Danny is seen in his blue dress shirt. However Jamie who is a patrol officer and thus would wear a blue shirt like Danny, he is wearing a white shirt like Frank and Henry. White shirts are for members of the NYPD with the rank of Lieutenant or higher.
Danny is seen at the dinner table after the funeral in his Class A uniform with a 2-3 day beard. This would never happen as officers are required to look presentable in their Class A.
At the NYPD Cypress Hills cemetery, Frank mentions that "Henry Haywood, Motor Patrol" was killed as a member of Theodore Roosevelt's Rough Riders at the Battle of San Juan Hill, which occurred in July 1898. Haywood was actually a MOUNTED (Horse) Patrolman (consistent with being a Rough Rider), and the NYPD did not form its first motorcycle unit until 1911 and its first patrol car unit until 1919.
Eddie is said to have just graduated from the academy and being partnered with Jamie who has 3 years on the job. A rookie is always partnered with a much more senior officer that would have at least 5 years on the job.
In a scene at an officer's funeral outside the church, Jamie and Eddie are seen standing together, both in Class A's, but Eddie is wearing a white shirt.
In the take down scene at Grand Central, protocol would require more than two lone detectives to arrest a subject. In such a large area there is too many exits and space to operate (even assuming they had support from the transit police - who are not pictured, which is another anomaly). Taking down a subject in such a space would require time, coordination and manpower.