I'm borrowing this post for a stealth theory update on Amuro Tohru later. Please otherwise ignore this.
Amuro's Remaining Mysteries
V1.3 (January 6, 2014)
V1.2 (July 20, 2012)
Index:
Amuro's youth
Amuro's reason for hating Akai
Amuro's promise with Vermouth
Relationship with Date Wataru
Amuro's youth
Some of the Black Organization agents seemed to have been groomed for their eventual roles as children or teens, like Shiho. Amuro Tohru was also likely raised around Black Organization, although he may not have directly been a part of it in the beginning.
He was allowed to play tennis in middle school (~ages 12-15), assuming Amuro is not lying, so unlike Shiho his life wasn't completely centered on detective training. Gosho's weekly comment joked about the Org having a tennis club, which implies that he was around the Organization at this time. Amuro reported meeting Akemi, Elena, and Miyano Atsushi before they died, and if it is assumed that the Miyano parents died ~18 years ago, Amuro would be 11 when he met them. The Miyano parents didn't seem to go on family visits to outsiders often because the Black Organization would put whoever they visited under observation, (Hidden Bathroom secret) so Amuro being connected to the Org would make such a meeting easier. Amuro seemed to be especially taken by Elena, enough so that it was the first thing he mentioned upon meeting Shiho. It's unknown whether Amuro has any relations to other agents in the Black Organization (none seen so far have darker skin like him), or he was brought in from the outside because he looked promising.
Whatever the case, Amuro probably went to police academy on the Black Organization's orders to gain skills as an investigator and then began working for the BO afterwards because he cut off communication with Date.
In the January 4, 2014 interview with Gosho, it was revealed that Amuro's childhood will appear in the storyline as well as his childhood nickname. Amuro's dark skin color has a meaning. I imagine this will have to do with his induction into the Organization as a child, or his childhood in the Organization.
Amuro's reason for hating Akai
-Unknown, since the rivalry over Akemi was jossed-
Akai and Amuro having a rivalry over Akemi has been officially jossed by Gosho in the January 4, 2014 interview My feeling is that Amuro and Akemi were once friends, either in a relationship, or even childhood friends who grew up in the Organization together.
Amuro's promise with Vermouth
In chapter 800, Vermouth reveals she has a promise with Bourbon which went unstated at the time. Amuro's promise with Vermouth is most likely something along the lines of "do not harm Kogoro, Ran, and Conan during the hunt for Akai." There may be a second part: "do what is feasible to protect Conan and Ran from danger", but it is possible that Amuro is choosing to do this on his own for reasons stated below.
During the scene where Vermouth mentions that she and Bourbon have a promise with one another, Vermouth is was watching an accident scene where Amuro was finishing up a case that involved Conan being kidnapped and held at gunpoint. Amuro had just rescued Conan using a dangerous car maneuver. In response to Amuro's actions, Vermouth said Bourbon earned some trust and reminds him of the promise. We know that Vermouth has a soft spot for Shinichi and Ran. Watching Amuro intentionally have his car rammed with Ran inside in order to crash the car Conan is in may have prompted Vermouth to remind him of the promise because what Amuro did was dangerous -- and also praise him for earning her trust for a successful rescue.
Amuro has a history of getting Conan out of trouble. The first instance was during the Teito Bank Heist where Amuro (as Scar Akai) rescued Conan from a hostage taker by shooting him. Shooting the gun risked calling attention to himself. Amuro later crashed his car to stop Conan's hostage taker (although he was slow to start looking for Conan even though he suspected Conan was taken hostage). Next, he tried to warn Conan about an incoming tennis racket. When Conan was hit anyway, Amuro rushed over, pushing Ran out of the way to get to him and then doctored him. He also seems to want to follow along with what Conan is doing; he hacked into the Mouri Detective agency computer expecting pictures sent by the Detective Boys for Kogoro.
Jossed material related to Akemi rivalry motive
Relationship with Date Wataru
Awaiting more evidence...
Date and Amuro likely became friends because Amuro was in police academy with him. (I imagine Amuro went to the police academy under orders of the Black Organization to shape him into an investigator/detective type.) Date probably was a good influence on Amuro. Amuro cut off communication with him after graduating to avoid endangering him because the Black Organization doesn't like police and would be suspicious of Amuro continuing contact.