James Gordon said he only does Jack's dirty work with his left hand. He says that nuns beat him so that he would be right hand dominant. Given the religious undertones presented in the film, it makes sense given that it was a common belief for centuries that lefties were connected to Satanism and darkness.
Before Batman vanishes unexpectedly, Gordon starts to warn him not to show up at the house again. Seeing as Batman discovers the secret room and learns Gordon's secret within, it would make sense that Gordon is trying to cover his trail.
At one point, Jack's comments about diseased whores strongly implies he had first-hand experience with STDs. The primary STD of the era was syphilis, which is known to attack the brain (neurosyphilis) ... and at that time, the only treatment available was mercury compounds, which also can mess up someone's brain (and didn't work well against syphilis anyway). This along with his experiences in the war, explains why Jack turned psychotic.
Mercury poisoning causing insanity is known as erethism, aka mad hatter disease. And the Mad Hatter in recent times has been reimagined as a predator of women.
Sister Leslie's church is named for Saint Caedwalla, the Patron Saint of (repentant) Serial Killers. How fitting for a film featuring history's foremost serial killer.
Alfred doesn't think using fingerprints will catch on in law enforcement. He might have predicted the problem of falsely planted fingerprints and safeguards against them (i.e. gloves).
There's a reason why Gordon would murder his own children. If he had any daughters, he killed them to prevent them from growing up to be "whores". If he had any sons, he prevented them from sleeping with any whores like he did or from fathering "whores" themselves. The reason he burned them was to "cleanse" them of any sins so they would ascend into Heaven. Also more horrible knowing that any daughters may have been the Barbara who would have become Batgirl. However, if any sons was James Jr., then he may have done Gotham a favor.
He had children?
I could have swore I heard him mentioning he did.
During The Reveal Barbra says he was always "Gentle" with "Us" suggesting they had at least 1 child.
Actually, Barbara tells him earlier that he should be quiet because the children are sleeping. So there are at least two and, unless she is delusional, still alive at that point.
Given Barbra's own acquired mental illness and behavior that does strongly sugguest a delusional disorder, the fact that the children are never actually seen or heard on-screen, and the line about Jim burning the "sin" out of them, it's really anyone's guess.
