Sentence two is not possible in English - the sentence is wrong. "The doctor thought (simple past tense) I'd be better by today," or "The doctor thinks that I will be better by the weekend." In both sentences "by," means that you will be better on or before that point in time. For the first sentence, 'a,' is correct. If you want to use 'any,' then you can do it like this: "I don't believe any of what you are saying."

I think it is correct . "You can also say I don't believe any word of what you are saying"

"I don't believe a word you are saying" is the correct phrase. It would not be totally grammatically incorrect to say "any", and people would understand what you mean, but this is a fixed phrase, we always say "a word". If you want to emphasize it, you can say "I don't believe a single word you're saying" ＝ 你说的每一个词我都不相信。 "by the weekend" = 周末以前