Bruce asks Earle "Who's buying?" in regards to who is purchasing the now-public Wayne Enterprise shares, despite knowing full well that he himself is the main buyer. Earle though, only gives a vague answer, dismissing it as "a bit technical", as if he thinks Bruce probably doesn't know enough about business to understand. Thing is, since Earle doesn't know that Bruce is the main buyer, that means he isn't even paying attention to who's buying, which means the "it's a bit technical" part is really him changing the subject away from something he should know more about but doesn't...which means that Bruce is asking him a test or a tease to see how much he knew, which Earle fails. Which puts Bruce firing Earle and replacing him with Fox in a new light as well. It's not just that he likes and respects Fox as a friend and a businessman; it's also that Earle is genuinely incompetent or at the very least, not especially devoted to the company he's running so much as the money and status it brings him. And if you are the majority shareholder of that company, and it's a company that belongs to your family, you don't want someone who is not truly devoted to it to be running the thing.
At first, it seems unusual in the flashback to after Bruce's parents died, Gordon and the other police officers were wearing sky-blue uniforms, but cops in the present-day scenes wear black uniforms. But given that the Gotham Police Department is clearly based off the NYPD (up to the paint job used on police cars), it likely has the same history. Bruce's parents died in the mid to late 1980s/early 1990s, when the NYPD used sky blue uniforms, which changed to black uniforms around the late 1990s.
The color change also correlates nicely with Bruce's loss of innocence, as Gotham suddenly becomes a much darker and scarier place, and the police change to match. Same goes to the police car paint job: it's a 1990s paint job the NYPD used, but it uses a deep blue whereas the NYPD used a light blue.
Much is made in the film of Bruce trying to live up to his Father's legacy, and whether Thomas Wayne was making things worse in the long run by trying to help the people of Gotham instead of letting the city's cancer consume it and destroy everything. Ultimately, Bruce destroys the monorail his father built (to help the people of Gotham) in order to save the city his father loved.
On that note, in the prologue, Thomas mentions that he had the monorail built to help the people of Gotham. Later on, when Rachel is taking the monorail home from work, she is accosted by thugs, representing the city's further decay in the wake of Thomas Wayne's death.
Bat-armor material: Lucius Fox describes the body armor that becomes the Batsuit as "Nomex survival suit for advanced infantry." Later, Batman survives being set on fire by Scarecrow and is bruised from the fall but not burned. This might be an action-movie handwave, but what is Nomex? It's what the fire suits for racing drivers are made from.
It seems odd that Jonathan Crane was able to get on a panicked horse and make it run through very narrow streets and rear up for him. It initially seems like the filmmakers just wanted to throw in a version of the iconic comics image of Scarecrow on a black horse and were city boys who had absolutely no idea about horse's behavior. However, in Scarecrow: Year One, Crane grew up on a farm in Georgia. No actual horses, but it's possible he would have learned to ride. The filmmakers for Batman Begins probably knew exactly how damn hard it would be to get a horse to do that in that situation - they were hinting that it wasn't Crane's first rodeo.
Also, it's implied that he didn't take just any horse, it was a police mount from one of the officers sent to the Narrows. In all likelihood, that horse was trained to be at least somewhat calmer in chaotic conditions.
In Begins, the incident with the League was probably what caused Batman to start his no-killing policy. Before, he had no problem trying to kill bad guys, like Joe Chill, but when presented with an uninterrupted chance to kill a murderer in cold blood, he choked, and bought down the entire building trying to escape. That's not a plot hole, that's character development. Also, Bruce Wayne may have killed people, if in a combination of panic and self-defense, but Batman doesn't.
Listen to the notes that have to be played on the piano to open the Batcave in Batman Begins. Unless you're completely tone deaf, you'll notice that they're horribly discordant, do not work next to each other in that order and never could under any circumstances. Basically, they're horrible. The Fridge Brilliance is that since they're so incompatible musically, it's highly unlikely that anyone will ever play those notes in that order except to open the Batcave, preventing the Batcave from being opened by accident while someone plays the piano or whatever.
What little criticism Batman Begins received was about the jump-cut-heavy fight scenes, which made the action a bit too frenetic for some. Fridge Brilliance: Nolan stated he had a pragmatic reason for this, wanting to show Batman as a scarily fast attacker where the targets could not make heads or tails what they were being overwhelmed with. Nevertheless, he used more tracking shots for The Dark Knight's action scenes.
Wayne Manor is a metaphor for the legacy of Thomas Wayne. Near the start of the film, Bruce is considering killing Chill to avenge his parents, which goes against what his father stood for, and will probably result in him being thrown in jail, preventing him from pursuing his father's work. At this point, he tells Alfred that if he had his way, he'd tear the Manor down, brick by brick. Later, when the League of Shadows comes to destroy Gotham, rendering Thomas and Martha Wayne's efforts to save the city useless, they burn Wayne Manor down. Finally, at the end, Bruce has saved Gotham and decided to become its protector, continuing his father's work. Thus, he rebuilds Wayne Manor, brick by brick.
The League of Shadows isn't just an Expy of The League of Assassins/The Demon from the comics. It's also an Evil Counterpart to the Justice League. Bruce Wayne was the Token Good Teammate for the LoS, but is frequently depicted as the Token Evil Teammate for the JL. The LoS members all wear black uniforms and share the same skillset, while the JL members all wear different multicoloured outfits and have unique abilities. The LoSwants to destroy corruption, while the JL wants to empower and repair. The LoS operates in secret, using underhanded tactics like weaponizing diseases and economics to get the job done, while the JL operates out in the open, using its members' powers. Both Leagues also had to deal with their share of stubborn enemies that they have to keep fighting (The JL has an assortment of super villains, while the LoS has Gotham) and the leaders of both go up against Batman.
As part of his initiation to the League of Shadows, Bruce is told to execute a man they say is guilty of murder. The League don't present any evidence of his guilt—it is a test of loyalty and of not questioning their authority. In other words, it is entirely possible that they just abducted an innocent man and told Bruce he is a murderer; worst still, they might even have murdered the victim in question and framed him for the crime for all we know. The real horror is how much sense this makes- if part of the initiation into the League is executing a (kidnapped) prisoner, then every member of the League must have done this or something similar, and if that is the case, the odds of them "just happening" to come across a guilty person every single time they are about to initiate a new member are staggeringly low.
Given that the League believes in collective guilt, they'd be just fine with the above. Moreover, all of the films' League members are on suicide missions to destroy Gotham. Clearly, they don't believe in their own innocence, either.
There's good reason to suspect that the LoS hated Thomas Wayne. A monorail doesn't seem like a big deal, but real life studies have shown that one of the biggest contributing factors to wealth redistribution is the availability of public transportation. Basically, poor people that live within a mile of public transport are much more likely to get jobs and get out of poverty. Thomas Wayne probably really did bring most of Gotham's poor out of poverty and might've saved Gotham's economy as well. No wonder Ra's hated him...
Remember the part from the chase scene when Batman is driving the Tumbler over the rooftops with Rachel inside? Got it? Now remember that Rachel was still under the influence of the Scarecrow's literal Nightmare Fuel and imagine what that alreadyscary ride must have looked like.