On average, how literate in Mandarin are overseas Chinese? I'm hoping to be able to provide better service to some Chinese customers I occasionally see at my night job (hardware store) who have difficulty describing what they're looking for in English. I can't be sure what variety they speak (and if they speak a regional dialect of Mandarin, I would still be up a creek), and anecdotally, I gather that some Chinese immigrants in the US do not speak Mandarin fluently.

From what I understand, one of the difficulties of studying non-Mandarin languages such as Yue, Wu, or Min Nan is the fact that speakers of those varieties, from having to read Mandarin in everyday life, simply read and write Mandarin while speaking their own dialect.

Is this the common experience among Chinese immigrants? If so, I think I could focus particularly on reading and writing, which I'm already better at than speaking and listening, and have more of a guarantee of communicating effectively.
For any non-mandarin speaker that doesn't understand the Mandarin culture, this is a common confusion.

This is the unique characteristic of Mandarin language: in Chinese culture, people tend to communicate through the Chinese writing, instead of speaking. The "Pu-tong-Hua 普通话"(China) or "华语“（Taiwan）is indeed a modern language that evolves from Beijing dialect with a history of no more than 200 years. While the official speech used by bureaucrats is called "官话”. But intellectuals are still communicating with each other using the Glyph. 

Because China enters imperialism era way ahead of Europe (at least 1000 years), there is little need for "nationalism", i.e. to promote a single language nationwide.  Nevertheless, the nationalism process is on the way. It is no different than many countries that teach "national language" together with localized dialect. This takes time.

In fact, many countries still maintain localized dialect. However, phonetic language cannot retain the same literacy level of Chinese glyph, thus localized dialects are still confusing for other languages.   Nevertheless, localized regional terms still appear in various Chinese dialects that enter the language. 

(update)
There is a book call  说文解字(shuo wen jie zhi) (written around year 100-121), which selects a number of words that user by various region. 

The online zdic 汉典 contains reference to some localized dialect, under the section 音韵方言