AFAIK brackets are for when something in the quote is changed. Like of the actual quote is “he is a very smart person” but you wouldn’t know who “he” is from the context, you would change it to “[Cecil Adams] is a very smart person”
I’m sure someone with a style guide will be along to give a concrete lesson, but my understanding is A and B are not used at all in any major styles, and C is only used in some. Another option is “The dog […] has many fleas.”
At least in legal citation, brackets with nothing in them are appended to the end of words to make the subject/verb agreement or the singular/plural nature of the word compatible with the new sentence. For example, take the following sentence:
It is clear that the plaintiff makes a mountain out of a molehill.
If I was quoting that sentence in a context in which there was more than one party to which I was referring, I might say:
Here, the defendants “make a mountain out of a molehill.”
The only use I’ve seen of are what Aioua posted and to indicate an omission from a quotation [. . .]. The most recent MLA guidelines say that you should use ellipses to indicate something omitted from a quote, and only use brackets when there are ellipses in the original text and the reader could be confused as to which were added and which were already there. So if the original text is:
“I . . . would like some ham and cheese sandwiches for lunch.”
The quoted text could look like;
“I . . . would like some [. . .] cheese sandwiches for lunch.”
In legal writing, original ellipses are generally omitted (and a parenthetical is added indicating their omission), so any ellipses in a direct quotation may be assumed to have been added by the quoter as a way of delineating omitted text. Brackets are used solely to replace words with other words (as Aioua says) and in the very limited instance I describe.
I do it the other way because it retains as much as possible of the original quotation, thus indicating to the reader that “make” is being substituted for “makes,” and not some other word like “fashions” or “creates.”
Normal procedure for two-layer parenthetics is to use square brackets for the outer parentheses and curve parentheses for the inner set. [This would be a (more or less) good example.]
Substitution of verb forms nearly always calls for bracketing the substitute word, e.g., [make] or [made] in place of an original with “makes.” On occasion, a regular third-singular present or a regular past tense will be marked by simply bracketing the added [s] or [ed].
If you modify a direct quotation in any way other than omission, the change should be bracketed. “He was assassinated by John Wilkes Booth” can be clarified out of the context of the short biography of Lincoln it was lifted from by either “[Lincoln] was assassinated by John Wilkes Booth” or “He [Lincoln] was assassinated by John Wilkes Booth” – using either substitution or clarifying bracketed appositive. An omission should be marked by ellipses supplied by yourself as quoter, and normal English style does not bracket them, though some style manuals apparently call for it. (The one exception to this principle I might see is where the author’s style included the use of ellipses and you as quoter also supply further ellipses, as clarity to the reader which ellipses are auctorial and which editorial.
An ellipsis is always three periods in close sequence, followed by whatever punctuation is appropriate from the original, unless superseded by the shortened quotation’s use within your sentence. Hence if you are closing out a sentence with a quotation that you have clipped before its end, you will end the sentence with four close-set periods and a close-quotation mark, three for the ellipsis and one for the sentence-ending period. For some reason this is always confusing to people. Even though they would have no problem with, “Wherefore art thou…?” or “Cry havoc…!” the idea confuses them that a sentence should end “to have and to hold, for better, for worse…” In each case the closing elements of the sentence are three periods for an ellipsis, a sentence-closing punctuation mark, and a close-quotation mark. The fact that in the last example the sentence-closing mark was a period matching the three present in the ellipsis tends to cause cerebral meltdowns.