Question: I searched a lot and kept expecting to see this discussed: Why doesn't Gordon arrest Ramirez after the hospital explosion? At this point he's done a mea culpa to Dent re not taking his advice on MCU corruption; he's received a trusted text message about Ramirez & Berg; and he's experienced Berg's betrayal first-hand. Yet he talks about Dent being missing in front of her, then entrusts her with critical operational duties. Even though they're close and in disaster conditions, his utter failure to call her out on anything is bizarre given what's already happened.
Answer: At this point, what proof does Gordon have? The text message that you speak of points out Ramirez and Berg as people who have relatives in the hospital, nothing more. Berg wasn't necessarily a crooked cop, he was just pushed to the edge when he tried to kill Reese because he wanted to ensure his wife's safety. You can see in the moments leading up to it that he's having a really hard time with this. Coincidentally Ramirez's mother was indeed in the hospital, and Ramirez took bribes to help with the Medical bills. To my recollection, Gordon has nothing linking Ramirez to the explosion of the hospital.
Question: In the scene where Batman shows Lucius his giant sonar thing, why is he talking in his big scary voice? Lucius knows who he is, so why does he bother disguising his voice?
Answer: It's habit, and a sensible one at that. When he's in the mask, when he's being Batman, he uses the voice, even if the person he's with knows who he is. If he drops back to his normal voice with some people, it sets the precedent for using it while wearing the cowl, which means that he's more likely to slip up and use it around people who don't know, potentially revealing his true identity. If he sticks rigidly to using the voice when kitted up as Batman, regardless of situation, it minimises that possibility.
Answer: At the bottom of his cowl Batman has devices designed to keep his voice at that level. The director revealed it, of you look at the right screenshots you can see speakers.
Question: How was Batman able to get to Harvey Dent faster than police to Rachel? I mean, I can think Bat's technology being way faster of course. But for example: if all units were sent to Rachel's location, there might have been some units patrolling near her location so it would have been faster to have them go there first. Right? What other possibilities am I not considering?
Answer: I honestly think within the context of the movie, we are just meant to assume that most units are either at the police station (given a high-profile terrorist suspect - the Joker - and many of his goons were just captured) or still at the scene of the massive chase that just occurred earlier in the night, since there was a lot of destruction or chaos. Thus, there simply wouldn't be any available units closer to Rachel's location. It's a little far-fetched, sure... but it's basic "suspension of disbelief." It's required dramatically for the plot to progress. (Similar to the fact that in the sequel, many of the Gotham City PD got stuck underground by Bane... it's a little far-fetched, but it works for the plot.) Batman was simply able to get to Dent's location faster since he's using the batpod, which is a bit of a technological marvel and can get around much quicker and easier than a police car.
Answer: This could be a plot hole, but I would posit: Given how corrupt/easily corruptible the Gotham police are, it's likely the Joker may have bribed/threatened/otherwise coerced the beat cops to a) steer clear of the areas where he was holding Dent and Rachel, or b) ignore the call to go to that location. You're right that Batman's tech is what allows him to reach Dent faster than Gordon et al. Reach Rachel; the Joker set it up that way, so that whoever Batman saves means the other one dies.
Answer: Batman showed up only a few seconds before the GPD, they arrived as the building blew.
Question: Near the beginning, when Alfred sees Bruce is not in bed, he goes to the Wayne Enterprises private property. There a weirdly shaped metal structure that looks like it's tilting over and what looks like a dilapidated building in the middle tilted to the side. What is the structure supposed to be?
Answer: It looks like a large panel on the side of the huge crane. I think it has openings for access to the crane machinery or something like that. Maybe a crane operator could describe it better but that's what it looks like to me.
Question: Why doesn't the movie acknowledge the actress change for Rachel? I mean everyone who watched Batman Begins will be clueless if they didn't know that Katie Holmes chose not to reprise her role. But the movie never explains this and are people just supposed to go along with it?
Chosen answer: Actors are often changed between movies, occasionally with acknowledgment, more often not. James Bond immediately comes to mind, Jennifer in Back to the Future, Bruce Banner and Rhodey in the Marvel films, Clarice Starling in the Silence of the Lambs/Hannibal...the list goes on. The recasting of Evelyn in The Mummy: Tomb of the Dragon Emperor got a passing nod to the audience (actually, as did the first recasting of Bond), but most of the time audiences are just expected to accept the change and move on.
Answer: I'm really not sure how you think a movie can address an actor being replaced. Unless it's something like Deadpool or Wayne's World, where characters speak directly to the audience, there's no real practical way for a movie to openly acknowledge that one of its characters is being played by someone new.
Answer: To add to the other answer, Lau is likely to have heavy security no matter where he is, so an equally elaborate plan would be necessary even if he left his tower.