What does Sholem Aleichem's description and use of a luftmensch tell us about (these are just examples) either the rise of Yiddish culture during the turn of the 19th/20th century, Jewish studies as a whole, Eastern European Jewish culture, or Jewish literature? How are we to understand Menahem-Mendl and his adventures?
I couldn't help laughing when I saw the beginning of your question. The term "luftmensch" is certainly humorous, especially the way that Aleichem uses it and how OFTEN he uses it in his writing. (Don't get confused with the author's name, he was born as Rabinowiz but always went by Aleichem.) As a precursor to your real question, let's talk about this word's meaning. Aleichem, as the leading author of Yiddish literature, takes many of his words from the combination of German/Yiddish that can be found during the turn of the last century.
In this case, the word "luftmensch" comes directly from the German words for "air" and "person." An easy way to remember the meaning is that it is similar to what we would call an "airhead." There are a couple of specifics that we need to keep in mind about Aleichem's usage of it, though, especially where the character of Aleichem is concerned. In Yiddish, the term "luftmensch" is almost always used to refer to a male (as opposed to a female). Why? Because a luftmensch is someone who may be quite philosophically minded, but is horrible at maintaining a family and/or being the breadwinner of that family. A luftmensch, then, is completely lacking in common sense and, usually, achievement and materialistic success. The term has a negative connotation, often meaning the person is fairly useless. Here is my very favorite quote in regard to the Yiddish term in the question:
A Luftmensch is a good friend for holistic, unbiased and unpushy advice, and useless as a breadwinner.
As for the use of Yiddish terms in the book and in our culture, the use of words (and other words like it) show the influence of the Eastern European Jewish culture on America as a whole. Aleichem is considered a "founder of Yiddish literature," and as such became very popular as a result, mostly, of the humor involved. Partially as a result of his writings, we have adopted many Yiddish terms into our common speech even today. I can't help but think of Mike Myers on Saturday Night Live doing the character of Linda Richman during "Coffee Talk" and feeling all "verklempt." That is a perfect, lasting example.
In regard to how we are "to understand Menahem-Mendl and his adventures," my opinion is that we are to understand them with humor and respect for the Yiddish culture now found (and prominent in some places) in America. There is such humor in the exaggerations of Menahem-Mendl in his original letters to his wife as he builds up his life in the city and his exit from the traditional. There is also humor in his wife's inability to understand. We should also recognize the parallels of Menahem-Mendel's life and the author's life, such as his move to Kiev and the move to the fictional Kiev ("Yehupetz") as well as his immigration to the US and his character's consideration of that particular move.
In conclusion, we should thank Aleichem's contribution to literature through Yiddish culture. We have a lot to learn about and laugh at because of him.