GW, there are rules for these things (which unfortunately sometimes differ between AmE and BrE usage), and there is an over-riding guideline of clarity:  if it gets too messy or iffy,  reframe the sentence.  With regard to your specific examples:


The word 'and' can be used intermittently throughout the paper, but the paper may look mediocre. --This is correct, but they are not apostophes (which are used for contractions and possessives), but 'single quotation marks' (') as opposed to 'double quotation marks' (").  Some style manuals reserve single quotation marks only for quotes within quotes: She said, "I like your 'hot' pants".  Other style manuals allow you to use them instead of double quotation marks for single words or short phrases that are not direct quotations, but are indicating a special use of the word or phrase-- as you and I have done here with the word 'and'.

Do I like buying clothes from Walmart? Sometimes I do, sometimes I don't. This "sometimes-clause" is based on what I consider the ideal me and the other me. -- Here again I would use single quotation marks, because 'sometimes-clause' is just a new phrase introduced, not a direct quote; other style guides demand the double quotes.  Very particular stylists might even require  " 'sometimes'-clause".

I'm going to the store to buy that thing you wanted. You said it was "supervantabulous", but someone else said it was the dorkiest thing ever. -- You have punctuated this correctly.  If the punctuation is a part of the quotation, put it within the marks; if the punctuation is a part of the main sentence outside the quote, then the mark should go outside.  A notable exception (in British eyes) is the period at the end of a sentence.

This would be called my "sometimes-clause", and I haven't found a way to deal with it. -- This too is correctly punctuated (except for the possibility of  " 'sometimes'-clause")