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.
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.