Are there any differences in meaning between the following sentences? Are they equally usual?
A. He said not even a word.
B. He did not say even a word.
C. He did not even say a word.
I don't quite understand what you mean by "equally usual"; I don't think A and B are common usage. Out of the three, sentence C is the most idiomatic to me. When it comes to meaning, all three are attempting to say the same thing - He did not say a word. Only C gets it across correctly.
All of the following
"He did not even say a word", "He did not even look at me", "He did not even reply to my text message"
are more common constructions in comparison to the constructions of A and B.
I am having difficulty in finding a situation where B would be more appropriate than C.
"Even" is an adverb here. According to Cambridge, "even" is used to show that something is surprising, unusual, unexpected, or extreme.
Cambridge says:
When "even" refers to a whole clause or sentence, we usually put it in the normal mid position for adverbs, between the subject and the main verb, after the modal verb or [after the] first auxiliary verb, or after be as a main verb.
See Even.