Hi Matt - great questions!
Before I get into a response to them, however, let me pay you two quick compliments:
First, you've got the idea exactly right here, as your rephrased human/mammal and A+/study sentences are spot on! And that ability to recognize consistencies in meaning despite variations in appearance/phrasing is key on this test. So that's extremely encouraging!
Second, this degree of scrutiny and attention to detail—both in terms of linguistic subtleties and nuance (and minutiae), as well as in the book itself as you consider sidebar notes—is the hallmark of both big improvements and high scores. So keep that up and you'll find it really rewards you on test day
Now, to your questions about sufficient and necessary condition ordering. I think some of the confusion arises from unintended ambiguities in the words "before" and "after," as they can mean (at least) two potential things in this context: before/after in terms of sentence placement/construction; before/after in terms of real-life event timing (chronology).
What the book is referring to is the latter usage, where necessary conditions can happen at some point in time before the sufficient, or at some point in time after the sufficient. The "stairs" example illustrates this nicely, as the necessary condition "climb the stairs" must clearly occur prior to reaching the roof, and thus the necessary piece was the first event to actually take place!
Ditto the A+/study example: studying is necessary, but it occurs before someone takes the test (and certainly before they get a grade back).
So in the same way that sufficient and necessary conditions can appear in any order in a given sentence, so too can they occur in any order relative to the real-world passage of time.
The reason we make that explicit is because (1) people often (wrongly) assume that the condition that is given ahead of the arrow in the conditional relationship (the first piece) must have preceded the condition after the arrow, but that assumption is false and when used to construct conditional diagrams will get you into a lot of trouble! (sometimes people go in the opposite direction and mistakenly assign the first-occurring event as the sufficient condition, but it comes to the same thing), and (2) the other major reasoning type on the LSAT, Causality, has conditions that DO occur in a particular temporal sequence, since the cause must always precede the effect in order for the cause to make the effect happen. And since conditional reasoning and causal reasoning are so often confused, we highlight that particular distinction as one more way—and of course one more reason—to keep them separate from one another.
I hope that helps clear things up, but if you have any further questions by all means let me know and I'll do what I can to continue clarifying. Thanks!
Climb Stairs