In actually, the question of abbreviations next to periods is covered by style rules that are separate (although visually related) from those involving closing quotation marks and periods.
The Chicago Manual of Style (17th ed., 6.123) says this:
When an abbreviation or other expression that ends with a period occurs at the end a sentence, no additional period follows . . . Of course, when any other mark of punctuation is needed immediately after the period, both the period and the additional mark appear.
The study was funded by Mulvehill & Co.
Johnson et al., in How to Survive, describe such an ordeal.
Therefore, the correct uses of periods in your examples are:
The most common one is "vs."
Richard, Rich, Rick, etc.
Note that the question of the period in vs. before the final quotation mark is irrelevant in this specific situation, because the period here is actually part of the abbreviation; the terminal period is simply absent. (Or, in another sense, the existing single period is doing "double duty.")
As with other discussion around closing quotation marks and punctuation, an interesting note can be found in Chicago (6.13) regarding abbreviations with periods next to other periods—in particular, the third example:
When an entire independent sentence is enclosed in parentheses or square brackets, the period belongs inside the closing parenthesis or bracket. When matter in parentheses or brackets, even a grammatically complete sentence, is included within another sentence, the period belongs outside . . . Avoid enclosing more than one complete sentence within another sentence. In the third example, two periods are required—one for the abbreviation etc. and one for the sentence as a whole, outside the parentheses.
Fiorelli insisted on rewriting the paragraph. (His newfound ability to type was both a blessing and a curse.)
Felipe had left an angry message for Isadora on the mantel (she noticed it while glancing in the mirror).
His chilly demeanor gave him an affinity for the noble gases (helium, neon, etc.).