At 9:48 Alfred asks "Can (am I physically able to) I show you the door?". A proper English butler would use proper grammar "May (would you allow me) I show you the door?".
When Jim Gordon enters Barbara's apartment and discovers Selina Kyle, an open box of Frute Brute cereal can be seen on the table. This may have been intended to add to the series's timelessness; the original Fruit Brute (note the spelling) was discontinued in 1982, and was only reintroduced as Frute Brute in 2013.
The phobia session is supposed to be anonymous, bit when detective Harvey attends, everyone knows each others names, since anonymity is given due to the fact that such meetings withhold last names as well as the names of friends and family. Anonymity is further granted by the agreement of participants by not relaying information on what is said during such meetings to the outside world.
Starting at 36 minutes during the gunfight, both weapons fire seemingly unlimited bullets.