Other mistake: It is previously explained that Jake cannot grow a moustache, then a few seasons later Jake goes to prison and grows a moustache.
Other mistake: Holt takes the crown out of the back of Jake's cabinet. However this wouldn't be possible as the crown was in the top drawer. This would have a back on it and it wouldn't be accessible by just removing the back of the cabinet.
Other mistake: When Amy reaches down to get the key out of her shoe she literally only bends slightly. There is no way that she could've reached all the way down to her foot, or retrieve the key in the seconds before she gets it. (00:14:30)
Answer: In reality, the Commissioner is a civilian administrator, appointed by the mayor of New York City. He (as of this writing, the role has always been filled by a man; the show gets that part right) is usually someone who has risen through the ranks of the NYPD, but upon becoming Commissioner, is no longer a sworn member of the force (the highest sworn rank is that of Chief). Since Commissioners are appointed, technically anyone can serve in the position, including Deputy Commissioners (who are appointed, in turn, by the Commissioner himself).
Update: On 1 January 2022, Keechant Sewell became the first ever female commissioner of the NYPD.