1. I don't read the novel yet.

2. I cannot start to do the work yet.
3. I will not open the bottle yet.
4. I don't want to go there yet.
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Are they all grammatical?

1. I don't read the novel yet.

2. I cannot start to do the work yet.
3. I will not open the bottle yet.
4. I don't want to go there yet.

5. I didn't read the novel yet.
6. I couldn't start to do the work yet.
7. I would not open the bottle yet.
8. I didn't want to go there yet.
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Are they all grammatical?

#1 is incorrect; #5 is a bit awkward; all the others are fine.

To correct #1, you'd write, "I haven't read the novel yet."

#5 would be much better if you omitted "yet."

Yes, all of the sentences you provided are grammatical. They follow the basic structure of subject-verb-object, and they use the word "yet" to indicate that something has not happened or will not happen at the current time.