Problem is to defend such a claim IN COURT - the copyright has to be legally valid - and Mr. Peters copyright claim would fail on those grounds and the case would be dismissed as Mr. Peters would be unable to prove he had a valid claim (Hell, Axanar is TRYING to claim CBS/Paramount's copyright is not 'clearly defined' thus the case should be dismissed.)
Not necessarily. Rights revert to the script writers in Hollywood per WGA rules. This is how Harlan Ellison and Neil Gaiman can publish their scripts; they, not the studio, hold the rights.
That said, I'm doubtful that Axanar is covered by the WGA agreements.

My point is that while it's possible that a blogger or writer would prevail if they published parts of the script and Peters sued for copyright infringement, it's not a slam dunk and would probably be an enormous nuisance.