Shiwan stole Genghis' coffin and probably threw his remains out somewhere.
In one of the podcast reviews of the movie, I've heard another example of Fridge Brilliance: in the chinese restaurant scene Shiwan Khan wears a nice suit, very similar to Shadow's. That's because Shiwan Khan, being the Ying-Ko fanboy he is, tries to mimic him. And another moment: in the same scene bullets shot by Shadow and Shiwan Khan hit each other. Rule of Cool? Not only. They were trying to scan each other's minds, that's why their bullets ended up hitting each other.
When Dr. Reinhardt Lane is disarming the bomb, he mistakenly pulls the red wire. However, it works for one very good reason. He's color-blind, so the WRONG wire is RIGHT.
Even more fridge brilliance: In electronics, the red color indicates the right-hand side, because they both start with R.
Jim Harmon, in The Great Radio Heroes, had this to say about the Shadow's friend and companion:
The only other woman I can think of who has had to endure so much for a man is Lois Lane, perennial girl friend of Superman. With such a similarity in name and character, can there be any doubt that these two girls, so attracted to men who wear capes and mistreat them so, are sisters?
Shiwan constantly talks about being "the last descendant of Genghis Khan", but Genghis Khan was actually a prolific breeder in his lifetime and had potentially hundreds of descendants. However, even allowing for the limitations of science at the time the film was set, it would certainly be in character for a man as arrogant as Shiwan to still regard himself as the "only" descendant, either because he believes he can trace a direct line of ancestors all the way back to Genghis Khan or just because he's simply that crazy.
If Bronzium is believed to be a legend, how does Dr. Tam know exactly how to test for it, what it is capable of, and what would be needed to build an atomic bomb with it? Not only that, how would either Dr. Lane or Dr. Tam know anything about building an atomic bomb twenty years before Einstein worked out the theoretical framework explaining why/how it would work?
As for how Tam could test for it, he probably conducted a serious of simpler tests before what we saw in the film directly that allowed him to determine what it wasn't, at which point they could start to speculate about more theoretical metals.
Why was the inscription of Genghis Khan's coffin written in Latin? His empire never reached most of the nations where Latin was considered an important language. Shouldn't the inscription have been in Mongolian?
Most likely Shiwan's sense of 'updating' his ancestor's history and role.
Bronzium is probably Uranium or some other naturally-occurring substance that can be used to make an atomic bomb. Lamont and Dr. Tam spend a lot of time handling it with their bare hands and carrying it in their pockets.
Lamont's coin seemed to be old enough to no longer be radioactive. At any rate, Shiwan carried it for a pretty long time with no ill effects.
Uranium emits alpha particles, which interact strongly with matter but are easily blocked. The layer of dead skin on the body is enough to shield us; the risks come from internal exposure, direct to living tissue.
Also, keep in mind that Shiwan wouldn't want to kill Lamont off in that manner, given his fixation on the other man as his 'worthy' counterpart.
