Some questions from forall x I was doing practice exercises of chapter-3 of the textbook forall x: Calgary An Introduction to Formal Logic.
There are some questions confusing me (there answers are not given in the solution booklet):

B. For each of the following: Is it a necessary truth, a necessary falsehood, or contingent?

(3) If wood were a good building material, it would be useful for building things.
(5) If gerbils were mammals they would nurse their young.



D. Which of the following pairs of sentences are necessarily equivalent?

(2) Thelonious Monk played gigs with John Coltrane. John Coltrane played gigs with Thelonious Monk.


Now, for the questions, I think (5) is a necessary truth because of the meaning of 'mammals' (but not 100% sure).
But, I am not able to decide for (3), which feels too ambiguous (e.g., good building material for what? Can there be good building material for something which is not counted in our notion of 'things').
Also, the pair in (2) should be equivalent unless there can be a primary and a secondary role related to 'playing gigs', which I am not sure of.
So, what are the right answers to these questions? Please explain.
 A: pg.18

For the rest of this section, we’ll take cases in the sense of
conceivable scenario, i.e., in the sense in which we used them to
define conceptual validity.... if we use a different idea of what counts
as a “case” we will get different notions. And as logicians we will,
eventually, consider a more permissive definition of case than we
do here.

pg. 21


*

*An argument is valid if there is no case where the premises are all    true and the conclusion is not; it is invalid otherwise.

*A necessary truth is a sentence that is true in every case.

*A necessary falsehood is a sentence that is false in every case.

*A contingent sentence is neither a necessary truth nor a necessary falsehood; a sentence that is true in some case and false in some other case.

*Two sentences are necessarily equivalent if, in every case, they    are both true or both false.


I assume that here, “necessary truth” means logical truth, “necessary falsehood” means logical falsehood, “necessary equivalence” means logical equivalence, and “valid argument” means an argument whose conclusion is a logical consequence of its premises.
As such, I'd say that

*

*(3) $\;[P\rightarrow P]\;$ is a tautological, therefore necessary,
truth;

*(5) $\;[P\rightarrow Q]\;$ is a contingent sentence that is (synthetically) true;

*(2) $\;[G(x,y)\leftrightarrow G(y,x)]\;$ is a contingent sentence
that is (analytically) true. (Whether the predicate/relation $G$ is symmetric depends on interpretation.) So, the pair of sentences is not “necessarily equivalent”.

