grammar - Do reverse clauses with 'be/have/do' require 'that'?

I am looking at these sentences:

1. *The fool he is said this.*


2. *The job he has is hard.*


3. *The job he does is hard.*



4. *The suit he wears is black.*

The reverse clause #1 seems completely wrong to me. #2 and #3 look somewhat questionable, and #4 looks completely correct to me. However, adding that (The fool that he is said this.) seems to make it more correct in each case.

It seems like I would always prefer to use that for all be/have/do-based reverse clauses to make them right.

My questions are: Is there a rule that makes #1, #2, #3 wrong? Or, perhaps, they are all correct?

On a side note, I have never come across a book that would comprehensively describe and explain English grammar on a formal level. Books tend to go by example, and kind-of coach people to do it right only on an intuitive level.