Everything you say is correct.

When you ask if the sentence is "true," do you mean "is it correct?"  Yes, it's correct.

<< What they share in common, however, is the extraordinary variety of plant and animal life-forms that are a necessary part of the ongoing process of their formation. >>

I think your book is wrong in claiming the word "that" qualifies "variety."

For one thing, the word "that" can't qualify anything.  As a relative pronoun, it stands for, or represents a word or phrase.  In my opinion, it stands for "life-forms."  "Life-forms" is the antecedent of "that," or the word which comes before it, which "that" represents.

"That" is the subject of a relative clause, which runs from "that" to the end of the sentence.  The clause, taken is a whole, is what qualifies the antecedent, "life-forms."

The subject of the sentence is "what," or the whole phrase, "What they share in common, however,."

"Variety" is a very important and key word in your rather complicated sentence.  It's the noun complement of the main verb, "is."  Everything else qualifies "variety" either directly or indirectly.

"Extraordinary" qualifies it as a simple adjective.

The prepositional phrase, "of plant and animal life-forms" qualifies it by answering "what variety?"

And, as I said, the rest of the sentence (the "that" clause) qualifies "life-forms." 

So you can say that the "that" clause indirectly  qualifies "variety."

Notice that your sentence doesn't say that the variety is necessary.  It says that the life-forms are necessary.  This is (I think) where your numbers question comes in:

Do we mean   -   The variety that is  necessary; or the life-forms that are  necessary?

The relative pronouns "that" and "which" can be either singular or plural. If the antecedent is singular, then the verb  must be singular.  If the antecedent is plural, then the verb  must be plural.