There are multiple errors in the sentence. Below are two grammatically correct possibilities. The second sentence would be preferable in formal written English.
The writers and chapters each has contributed to are as follows.
The writers and chapters to which each has contributed are as follows.
To break it down a little, the subject of the sentence is writers and chapters. The verb are is present tense, and so as follows should also be present tense.
Each has contributed to and to which each has contributed are present-perfect phrases that modify chapters. In formal English, ending a phrase (and especially a sentence) with a preposition like upon or with is frowned upon. This is not a rule to be messed with. You can read more about it here: http://grammar.yourdictionary.com/parts-of-speech/prepositions/Ending-a-Sentence-with-a-Preposition.html
To have to X means to be required or forced to do X. It's often equivalent to must X.
I have to go now = I am required to go now, I must go now.
One meaning of to have to be X is to be required or forced to have X completed or done.
I have to be finished by 2:00pm.
I have to be gone before John gets here.
Another meaning of to have to be X is to be required or forced that someone else does X to you. If there is no by phrase saying who, the default is by someone or by anyone.
She will have to be told to leave, otherwise she isn't leaving.
I have to be washed by the nurse since my arms are immobile.
My guessing that it is "-have to + passive voice-" combination here, where "-have to-" is that "kind-of-modal-verb-but-overly-not", am I right here?
Have isn't a modal. It can be a helping or auxillary verb as traditional grammar describes. Like do or be. I'm sure CEGL calls it something different. It's definitely not a modal because it's an option to use with modals. E.g. I could have gone and I could go are valid.
Your modals are: can, could, shall, should, will, would, may, might, must, ought and a few other not-so-common ones like dare.
But have in addition to being a helping/auxillary verb is also a standalone verb, and you have have X and a phrasal variation have to X.
E.g. this works:
I would have had to have gone to the park yesterday but my friend had gone instead.
can I rephrase this sentence in following way
Yep, because you are asking about the what the commands are, not really anything else about the commands such as whether or not she had to do them.
Best Answer
You're right, there is a tense problem. "Should already have exist" should be "should already have existed." This is the past progressive (or past continuous) tense combined with a modal verb "should." The auxiliary "have" cannot be combined with the present tense form of a verb: it always takes a past participle.
A more idiomatic way to say the same thing is:
But the original version is grammatical when "exist" is corrected to "existed."