This answer introduced me to Yiddish.
Is Yiddish taught and used in Germany/Austria/Switzerland/Lichtenstein? Is it a dialect of German or a separate but similar language?
I wouldn't call it a dialect like the Bavarian dialect. It is a High German language, derived from Middle High German combined with Hebrew, Aramaic and Slavic.
It was used by the Ashkenazi Jews ("German Jews") living in Germany along the Rhine and spread from there.
Yiddish is a separate language that split from German about 1000 years ago.
Western Yiddish was once spoken in the territories you mentioned. That dialect of Yiddish was however already moribund by the mid-twentieth century, as rates of assimilation of Jews to German-speaking culture were very high, and had been since the beginning of the Haskala in the eighteenth century. On the eve of the Holocaust, speakers of Western Yiddish numbered in the tens of thousands, versus some ten million for Eastern Yiddish. In peripheral areas such as Alsace and Switzerland, there may still be some vestiges of Western Yiddish in the form of traditional songs, and so on.
It is debatable whether Western Yiddish was a separate language from Eastern Yiddish, which is what all remaining native speakers speak, but mutual intelligibility was low in speech. Eastern Yiddish was traditionally spoken in Poland, Hungary, Romania, Soviet territories, etc., but now is mainly spoken in either the former Soviet Union or the diaspora (U.S., Canada, Argentina, etc.) and Israel.
Literary Yiddish, until around 1800, was based mainly on Western Yiddish, especially as spoken in Western Europe (e.g., in the Netherlands). Then, because of the rapidly declining numbers of Western speakers, the written language suddenly aligned itself with the vernacular Yiddish of the eastern areas.
Yiddish is now dying out among secular speakers, but is still an everyday community language for many Haredi Jews. In fact, there are thousands of children in Canada whose mother tongue is given as Yiddish in the census - at some point the increase due to the demographic growth of the Haredi population will start to balance out the losses of elderly secular speakers.
These days, there are second-language learners of Eastern Yiddish in Germany, as Eastern Yiddish is the only variety still spoken. I imagine that the majority are either Jews with roots outside Germany, in the original Eastern dialect area, or non-Jewish Germans with an interest in the language.