Jack mentions that his former colleague (and lover) "Diane" Hawthorne has been called to testify before the independent counsel. In a previous episode, Trophy (1996), her name was "Diana", not Diane.
McCoy says that the only way Purcell (the prostitute suspected of killing the victim) will testify before the grand jury is if Independent Council Dell offers her immunity from the murder charge, and if that happened the New York County DA's office couldn't touch her. However Dell is working on behalf of the Attorney General of the United States and convened a federal grand jury, meaning he can only give her immunity from being prosecuted for federal charges. The Attorney General, and Assistant U.S. Attorneys, can only grant immunity to federal charges, their immunity deal does not and cannot prevent a State Attorney General or a District Attorney from bringing charges against that person. Only a federal Circuit Court judge or a Supreme Court Justice can grant immunity from state charges, and only then under certain circumstances. Dell's offer of immunity would not apply to state charges, so she could still be prosecuted in New York state for murder.
When a Metropolitan Police Department of the District of Columbia (MPD) police officer pulls up Chesley Purcell's criminal record, the mugshot pictured is from the New York Police Department, not the MPD. Why would the MPD have an NYPD mugshot in their records? Even if the suspect was previously arrested by the NYPD, the MPD would have taken its own photograph of her when they arrested her. Police departments don't rely on one another for mugshots; each takes its own pictures whenever someone is arrested.
Janine McBride's murder would not be investigated by the NYPD as it is out of their jurisdiction. McBride was the Deputy Director of the Social Security Administration, which makes her an employee of the federal government. The FBI has jurisdiction over investigating violent felonies like armed robbery, kidnapping, rape and murder committed against federal government employees. It's possible the FBI might request the NYPD's assistance but the FBI would have the lead on the investigation. In addition she was killed in Maryland and her body was dumped in New York, to go from Baltimore to New York City (on the most direct route) would require traveling through five different states: Maryland, Delaware, Pennsylvania, New Jersey & New York. Whenever a person crosses state lines during the commission of a felony it becomes a federal case, with murder & kidnapping being the FBI's jurisdiction.