On the plans, here is a concise answer.
On the question of the Japanese — if we imagined that somehow those fallbacks didn't work and the Japanese found themselves with a semi-broken but not exploded nuclear weapon, could they do much with it or recognize it? If we are talking about pre-Hiroshima I think it is not impossible that someone competent (like one of their own scientists who was tasked with researching fission) could have figured out what it was if they thought to do the right kind of tests (e.g. to figure out that it was uranium or plutonium at the cores of these things). But it would not have been obvious, particularly to someone versed in only conventional military problems. The Japanese did have competent scientists but the question is whether they would have been the ones called to investigate such a thing.
After Hiroshima, of course, the fact that the US was dropping atomic bombs on them was rather blunt and obvious so if a weird bomb turned up, that might be a suspicion. Figuring out how it worked would take some time, even for something as relatively crude as the Little Boy bomb, because even it was jam-packed full of complicated fuzing electronics to make it work. The Fat Man bomb would have taken a lot of analysis to make sense of because it is not the obvious way of making a critical mass (using different velocity explosives to implode a solid core is not what comes first to mind — it is a pretty far-out concept).
If you are asking, could they have turned around and used it as a weapon immediately — I doubt it. Even with the unlikely assumption that they received these weapons undamaged, working out the fuzing circuitry (even for Little Boy) would have taken some time. With Fat Man it would have been quite un-obvious how you get it to work, without tracing through it all and figuring out how it is put together. And if they were damaged, it would have been pretty tough to salvage the material and get it in a working state. The odds would be much better with Little Boy than Fat Man. A broken Fat Man bomb is practically useless unless you are using the plutonium as a "dirty bomb" radiological dispersal device.