Wasn't it suspicious that Edward could be seen cavorting with Two Face during the launch party?
People were too busy panicking to notice. He could always pass it off by saying that he was trying to reason with him if anyone saw (which was technically true). Plus, the whole thing only lasted a few minutes.
Wasn't it suspicious that Bruce Wayne beat up all those goons in the circus?
As above, everyone was panicking and the whole thing was over in a few minutes. Plus, most of Gotham already sees Bruce in a pretty heroic light, and Bruce beating up some thugs in a life-or-death situation doesn't mean he is Batman.
Why was Robin sent to Bruce Wayne instead of a foster home? If Bruce is willing to donate his 10 million spare rooms to put up orphans and Gothan knows this, how come they haven't sent anyone to Bruce before? Knowing that Gotham's a crapsack town, Wayne manor would be an orphenarium at that rate.
It's Gotham, the city of the legendarily corrupt. Bruce "multi-billionaire" Wayne asks, multi-billionaire gets, no questions asked. He probably didn't even have to bribe anyone, just his status would be enough for the corrupt civil service to roll over and acquiesce.
The question was not that but why Bruce Wayne took in just that one orphan and not any other orphan. Is he regularly a source for the police to drop off wards of the state?
He did it because both he and Dick have now seen their parents murdered by a maniac in front of their eyes. Bruce probably gives a lot of money and support to orphans and charities, but he took in Dick because what happened there hit close to home.
It's implied, if not outright stated, that Bruce puts a fair amount of blame on himself, as Two-Face and Batman are rivals at this point, and Two-Face's actions that night, trying to flush out Batman, led to the Graysons' deaths. Taking it further, Bruce, as Batman, was unable to stop the mobster from throwing acid in Harvey Dent's face, so Bruce probably blames himself for Two-Face's existence in the first place. Taking Dick as his ward is a means of atoning for all that.
The bigger question is why he needed a new guardian. This version of Dick is clearly old enough to support himself.
I thought that was answered in the movie? He's supposed to be Under 21 and the law of whatever state Gotham is in is that kids under 21 are supposed to have a guardian, no exceptions.
Why did the security guards not mind being punched by the riddler?
Because if they didn't let Two-Face and Riddler punch them, they'd likely be shot and killed.
So it's completely normal for groups of kids to go trick-or-treating (without parental supervision) at Wayne Manor, even though it's on the outskirts of town?
Parents don't have to be hovering over their kids' shoulders the whole time they're trick-or-treating. I don't know about your neighborhood, but in mine the parents often stood well back from their kids.
In this tropers experience the presence of parents significantly hampers the feasibility of carrying out any nessecary tricks.
Besides, everyone knows rich people will have the most candy!
Not to mention this is Wayne Manor - justified or not, people would probably be more comfortable being there as the Wayne family were well known, well liked, and lacking in any sort of scandal. Given that this Bruce was also more aware of the need to be Bruce as well as Batman, he's probably also started to make efforts to avoid seeming like a complete idiot with no day job.
The problem is, Wayne Manor seems to be in the middle of nowhere. Even if the Manor grounds are entirely safe (as it happens, they aren't), it still looks like its a mile or two away from the rest of civilization.
Edward's apartment and cubicle is adorned with bobbleheads and fortune teller machines of a guy that's dressed in a green suit with question marks all over it; He even takes the jacket and hat from the fortune teller machine when he meets up with Two-Face for the first time. Who is this guy?
This one comes from earlier script drafts, but was lost in the final cut. In those drafts, this pixie-like figure was called the "Guesser", and he probably served as the Gotham Globe's puzzles-page mascot (it's never stated what his function was). Apparently, he was popular enough that merchandise was made in his image, and his likeness licensed out for fortune-telling animatronics.
Who the Hell puts a deaf man in charge of guarding a bank vault?
He wasn't deaf. He had something to help his hearing, but he clearly wasn't deaf.
How did Nygma make any money on the Box? Since he designed it while he was working at Wayne Enterprises, wouldn't WE have held the patent on the thing?.
He had just invented it and hadn't filed for a patent yet, and Bruce Wayne himself said that the company didn't want to get involved in brain-wave technology.
The Novelization has some lawyers bring this up to Bruce, but he says taking Nygma to court would look too much like sour grapes. And, though this is implied and not stated outright, he wants to give Nygma enough rope to hang himself.