The technique Nobusuke Tagomi uses to repair the cup is called kintsugi ("golden joinery") using lacquer mixed with gold dust. As a philosophy, it treats breakage and repair as part of the history of an object rather than something to disguise.
Nobusuke Tagomi's (Cary-Hiroyuki Tagawa) lack of memory of past events in the alt-world can be played off due to his life in the alt-world. His alt-self is an alcoholic and he can use this fact as a basis for memory loss.
The stethoscope the doctor uses in Hitler's death scene is an odd combination: a 1930's style bell and 1960's single tubing. This makes sense in the alternate reality of the series, since the modern stethoscope was designed by Dr. David Littman, an American Jewish doctor descendant of Ukranian emigrants. In the world of The Man in the High Castle, Littman would have been killed, purged or forced to flee, and his invention definitely not known in the German Reich.