Is this term only well-known in Southern Germany or also in the deep North? Is there short alternatives for modern High German? Does anyone know a similar English short expression?
The term isn't even well-known in all of southern Germany - a Bavarian, for example, might well not know it. According to Wikipedia:
Der Begriff wird insbesondere für einen Beigeschmack von Speisen und Getränken oder einen verdorbenen Geruch verwendet und im übertragenen Sinn für Sonderbarkeit, spezifische, anderen auffallende und widerwärtige oder lächerliche Art eines Individuums oder Standes benutzt.
The term seems to carry a connotation of an unpleasant or weird taste. I'm not aware of any single high German term with the exact same connotations. "Beigeschmack" probably comes closest, but isn't used in quite exactly the same way. In some cases, you can also use "anrüchig".
(Edited to add some suggestions, thanks, commenters!)
Actually "Gschmäckle" is being used and understood across Germany in its metaphorical use. There, it describes something about a person's actions that might possibly be regarded as inappropriate, especially in political or economical contexts.
E.g. Shortly after Chancellor Schröder was voted out of office, he took on a highly-paid position with the Russian state-owned oil and gas company "Gazprom". Previously (as Chancellor), Schröder had brought about a deal between Germany and Russia concerning the building of a new gas pipeline.
So, people said that Schröder's move to Gazprom had a 'Gschmäckle'.
In that context, "Beigeschmack" is a close alternative in High German.