Without knowing anything else beyond that some woman with associations with the FBI thinks it is a good reason to use Amuro on a case, it is possible that Amuro is the son of an FBI agent and was taken in by the Black Organization. Maybe the woman knew Amuro's parent and then incidentally came across Amuro as a detective and assumed that because Jodie was in Japan that he was over for the same reason and had followed in his parent's footsteps. If Conan makes the same leaps of logic, he could come to the conclusion Amuro is an embedded spy, but the conclusion is off because Conan doesn't know Amuro's history with the Black Organization. That's why Amuro tells Conan he is mistaken. I still don't think Amuro is an Akai because he was shown in association with the Black Organization (Elena) as a child and his skin color doesn't match the apparent middle brother's (or the rest of the Akai family who is pale).
Havin a connection with the FBI would open up a new avenue for why he hates Shuuichi. Maybe Shuuichi had some involvement in whatever situation led to Amuro being left behind and adopted by the Black Organization. (Death of parent? Lots of possibilities here.)
It seems that Conan understood a meaning behind the nickname "Zero" that we readers have not figured out yet. That seems to lead him to whatever conclusion he makes regarding Amuro's background.
It seems Amuro has a particular grudge against the FBI in general, and appears to be enjoying ribbing them. I think it is a stronger reaction than most members of the Black Org would have to the FBI. If Amuro does have FBI parents, maybe he does not know exactly what went down between them and the Organization, and blames the FBI (and maybe Shuuichi or the Akai family [maybe even daddy Akai]) for whatever happened. A confrontation with Shuuichi might clear some of that up since Amuro now knows the fake death trick. All that is left for Amuro now is to figure out Okiya is Akai, and Amuro already knows Okiya exists and cooperates with Conan from the cat + cold truck case.
I bet the solution to the case has to do with the fact that in America a check by the answer means correct, while in Japan that means incorrect. Circling an answer doesn't mean incorrect in America, but it seems like it is being used that way by the foreign teacher. The motive is probably something along the line of some diligent student commits suicide because he thinks he got a terrible grade and the teacher was mocking him with flowers.
I'll think about all this some more when there are more translations.