I haven't really had time to closely examine the case but I suspect the solution has something to do with the color of the tea.
For Amuro, the nickname "Zero" ("Rei" in Japanese) is another Gundam reference. The Gundam character Amuro Ray's given name is pronounced the same way zero (Rei) is in Japanese. I suspect there will be another real reason for Amuro's nickname in-story though.
Amuro seems pretty young when he met Elena, like 6-10 years old. (He really does seem to be a Black Organization child.) That would place his meeting with Elena back before Shiho was born, assuming she really is 18. It seems that Elena and Atsushi knew they were going away/dying before even Shiho was born. Given how close Amuro seems to be to the Miyano family, I wonder of that has something to do with the reason he hates Akai. Does Amuro know the whole story behind Akai's betrayal? Amuro didn't seem incredibly upset that Shiho apparently died though so I'm not really sure about that theory. We do know Amuro's skin color is relevant for some reason according to the interview with Gosho, so that is something to keep in mind.
Now that Amuro knows Conan met Rikumichi and that both Shuuichi and Rikumichi touched Conan's cellphone (from eavesdropping on Jodie) the only information Amuro is missing is knowing Conan has two identical cellphones and that he loaned one to Jodie in order to get the body in the truck identified. Those two pieces are all Amuro needs now to deduce how Akai faked his death.