Why don't German-speaking countries nowadays ever use "Fräulein" for a young woman instead "Frau"?
Is the word old-fashioned or obsolete? If so, why, and which decade after the 1940s did it really drop off the map in terms of writing, speech and as a salutation?
Yes, it is archaic, don't use it.
"Fräulein" was used to refer to unmarried women, opposed to married woman who would be called "Frau". This distinction was never made for men.
"Herrn" is just the accusative or dative form of "Herr" and not all related to "Fräulein".
Fräulein is not used anymore for an unmarried woman since about the 80s, and almost completely disappeared from usage by the 90s.
The process was started in 1954 when Elisabeth Lüders started the process in the Bundestag. By 1955, women had the right to be called "Frau" in official federal communication.
By 1972 the ministry of the interior decreed that "Fräulein" had to be dropped from official communication.
The 80s saw some debate in the population, but usage quickly declined. By the 90s, "Fräulein" was obsolete in West Germany. From what I gather, usage prevailed longer in East Germany, but quickly dwindled down after re-unification in 1990.
Wikipedia has a somewhat good overview over the usage of Fräulein where you can find sources for all claims made herein.
This is not at all a recent change, and "Fräulein" sounds indeed archaic for most of the population. You might sometimes hear it from very old persons.
That being said, the word "Fräulein" sometimes sees some usage when parents educate young children or maybe teenagers, when the fact that they are indeed not fully grown adults is highlighted with this word.
"Fräulein" is not used for a young woman, it is used to address or refer to an unmarried woman. It is equivalent to the English "Miss".
In contrast, "Frau" was used to address or refer to a married woman, equivalent to English "Mrs".
It is no longer common to make this distinction, therefore "Frau" is always used, whether the woman is married or not. Practical reasons are that it is not other people's business whether a woman is married or not, and guessing wrong can be awkward.
In contrast, "Herrn" is just dative or accusative of "Herr", often seen as part of the address on a letter: "(An) Herrn X" means "(To) Mr. X".
