Question: I have read many novels and found that the writers do not follow basic phrase structure rules while constructing a sentence. For example, they may write a sentence with only one word like "Fear" And sometimes, they may come up with a compound sentence made up of imperative and interrogative.

Is this allowed, or should writers follow accepted grammar rules?

Answer: 
It’s not necessary, but breaking conventional grammar rules can be a good tool depending on how it’s applied. 

For example, if a story is written in the first person and stream-of thought-narration is used, using sentence fragments or run-on sentences at certain points could give the readers a better understanding of the characters mental state, as long as it doesn’t confuse the readers. 

I am not sure whether your question is addressing only breaking the rules in narration parts, but another example of where writers might not pay attention to the rules is in dialogue. If a character doesn’t speak in proper grammar, the writer will likely convey that as well. Even if the character doesn’t have “poor grammar,” the writer will likely not have the characters speak in formal English because that isn’t vernacular in most areas.
When did grammar rules and 'phrase structure' rules become the same?

The correct answer is no, a writer does not have to follow perfect English per some English dept or professor, nor from any handbook that has no official status and of which there are many that do not all agree.

What the writer needs to do is be clear to the reader. And since you mentioned novelist it is far more likely to effectively use what you do not think is proper grammar.