These are called truncated sentences
It is considered informal to truncate sentences (removing words that are unnecessary while maintaining a sense of the meaning). Despite this, I'd like to point out that some of the words removed in your sentences are not correct. Sometimes truncation changes the meaning of the tense, too, with words like "had" being removed. I'll go through them one by one:
Thomas was not sure how it [had] happened.
Here, "had" simply specifies the past perfect tense and can be removed without changing the meaning. Since past perfect is somewhat implied and the specific tense doesn't matter too much, the word can be removed.
there was no proof they [had] even occurred.
This is the same story. The word "had" is used to show past perfect, and is not necessarily needed since the tense can be deduced.
He [had] not witnessed them-
Here, removing the "had" makes the sentence straight-up nonsensical. "He not witnessed them" is not grammatically correct, and the word "had" must be included for this to make sense. Replacing this with "He did not witness them" has a very subtle change in tense: He had not witnessed them means "At the time we are describing, the events had already happened and he had not witnessed them" but He did not witness them means "The events happened at the point in time which we are describing, and he did not witness them". Since the past perfect has already been implied, I would stick to "He had not witnessed them" for this one.
-but [had] lived in their aftermath.
Here, the word [had] can be left out, but it completely changes the meaning of the sentence. "But had lived" means that Thomas lived in the aftermath, but doesn't any more (past perfect). "But lived" is continuous. The reason for this is that the story is told in past tense. This means that "at the time that this story took place, Thomas was still living in the aftermath". Compare this to "at the time that this story took place, Thomas had lived in the aftermath but doesn't any more".