Punctuations for quotes in a list
Solution 1:
American style is to place the comma inside the quotes. This is universally the case in publishing and accords with all style guides (Chicago, AP, NYT, etc.). The only exception is is in academic works, particularly philosophy texts, where a word is being specially defined and offset with single quotes. That exception, however, is not widespread and some houses, such as Oxford University Press, use the single quote as closing punctuation.
Solution 2:
American usage is "yes," it should be inside the quotes even if it is not part of the quotation. British usage has it outside unless it is part of the quotation (as far as I know).
Note that it is becoming increasingly common in American usage to move the comma outside of the quote because there is a growing subset of the population who use string literals in programming and the comma is not part of the data. The "Comma in the Quotes" rule is still expected however.