I think that I now understand most of the puzzle.
I would guess that the answer to "In what city did he meet his emperor again?" should be
Vienna (= Wien).
Explanation:
The main theme are
Nobel prize winners of physics in odd-numbered years, starting from the year 1901;
this is indicated by "Every second year he wrote me a letter";
The puzzle covers 1901, 1903, 1905, 1907, 1909; hence the answer should fall into the year 1911.
He was born at an Imperial hospital.
Wilhelm Conrad Röntgen; Nobel prize for physics 1901
Imperial=Wilhelm (Emperor Wilhelm II); hospital=Röntgen (X-rays)
He had three friends. One of them founded Red Cross half-life.
Peter and Mary was curious about life.
Henri Bequerel, Pierre Curie, Marie Curie; Nobel prize for physics 1903
Red Cross = Henri (Henri Dunant); half-life = Bequerel;
Peter=Pierre; Mary=Marie; curious=Curie
Once he travelled with another friend to Africa, to commemorate the church founder, singing Hallelujah without 'o'!
Philipp Lenard; Nobel prize for physics 1905
The Apostle Philip is regarded as the founder of the African church.
Leonard Cohen sang "Hallelujah"; Leonard without 'o' = Lenard
Einstein's forefather plays golf.
Albert Abraham Michelson; Nobel prize for physics 1907
Einstein=Albert; forefather=Abraham; golf=Phil Mickelson
My friend's favourite object was a brown radio.
Karl Ferdinand Braun (brown); Nobel prize for physics 1909
Guglielmo Marconi (radio); Nobel prize for physics 1909
In what city did he meet his emperor again?
Wilhelm Wien, Nobel prize for physics 1911
German physicist; the first name Wilhelm is the name of the German emperor Wilhelm II, the family name Wien is the German name of the city Vienna.
My friend's birthday is in the 111th period
Wilhelm Conrad Röntgen; Nobel prize for physics 1901
The chemical element with number 111 is Roentgenium, called after Wilhelm Röntgen.