[RayChuang88]

If I remember correctly, the device tested at the Trinity site had the same fissile material shape as the Fat Man bomb, but they surrounded it with a lot of extra material and test sensors because nobody at Los Alamos knew if this device would actually work. But once the bomb was successfully tested on July 16, 1945, they could assemble Fat Man with assurances it would actually work.

[chimera]

Yes, the plutonium core was the same. That design remained the same until the latter 1940s when the first levitated core designs came into use. That led to the boosted fission weapons, which were the progenitors of the thermonuclear weapons of the early 1950s.