This is very interesting. Do you know any good references where I can read more about this perspective, as well as similar appearances of (co)monads?

Yes, please! In plain language search, Google and other search engines are quite good at finding relevant synonyms that you didn't type. Broadening a TeX-math search to allow for "math synonyms" or alternate notations would be just one of many useful features this technology could have.

I'm all for objections and devil's advocacy, but please...hold the straw men.

@user76284 I'm not saying math classes should devote a class period to climate-related topics. I am pushing back on the exaggerated claims of time costs. When teaching I take every opportunity for side remarks about relevance to other fields, as building mental networks of knowledge is my favorite method of learning and teaching. This takes very little time. If, further, one wanted to devote a whole class period to the relevant math of one's favorite topic, then in the grand scheme of things this is also not much time. Note: the relevant math...not classes on hunger, history, etc...duh!

@MonroeEskew "devote all our time"? You are catastrophizing. It is not time consuming to make occasional mention of various applications, or even to devote one whole class period. Other more relevant applied science courses should certainly discuss the connections, but from an educational standpoint is is very helpful to encounter the same information in multiple contexts and perspectives. Also, from my experience with students, math education really benefits from highlighting relevance to "the real world".

For reference, in Mac Lane, Moerdijk - Sheaves in Geometry and Logic (bottom of p75), there is a sheaf-theoretic description of smooth manifolds which does require the second-countable condition.

Ah ha, and here we have an clear and incontrovertible obstruction. Thank you, @NoamD.Elkies

@Jef Thank you. I don't fully understand your comment, but I am happy to upgrade hypotheses so that $\pi$ has a section. Is there anything you can say about that situation? Also, what is the method for showing the result for question 2 (locally)?

Just as a local ring is not by definition a subscheme, I don't think you could expect a subscheme (not finitary enough), but is the thing you're seeking something like inverting all functions whose zero locus is disjoint from $Z$? It seems like it would be something like the coproduct in the category of $X$-schemes of the local ring at $\eta_Z$ and the formal completion of $X$ at $Z$. When $Z$ is a section, I would be thinking about relative effective Cartier divisors instead of arbitrary functions. In any case, I myself am very much interested in seeing a good answer.