Yes, that was a kind of typo. Peikoff's "inductive proof of causality" is the subject under discussion. Yes, and by the way proof is also a method of integration because what is proved is related to other knowledge.
Yes, the fact that you can contemplate the axioms and relate them to each other is a form of integration even though Peikoff would deny there is proof or derivation or deduction happening. The order of Existence, Identity, and Consciousness has methodological (epistemological) significance in order to affirm Primacy of Existence and deny Primacy of Consciousness, but each is a mentally abstracted facet of existence which exhibits all three simultaneously.
Causality merely appears to come "after" Identity in that it is easier to understand or imagine some object as static and then add the dynamics but in reality everything that exists is always acting (even if slowly). Understanding Identity as static omits the greater part of an existent's Identity, how it acts.