The fact that various finiteness conditions lead to good theorems which are manifestly false in their absence seems like a good explanation of why they are important. (In fact, I am having trouble thinking of a wholly different kind of explanation for why anything in pure mathematics is important.)
I think you are on to something to the extent that we need to give nonexamples and counterexamples along with our theorems in order to give students even a fighting chance at appreciating them. In the realm of commutative algebra this was something that was notoriously underappreciated until relatively recently: I recall well Rota writing about the "hygienic theorems" [Rota, Indiscrete Thoughts, pp. 215-216] in algebra, e.g. things like "Every regular domain is normal". As he wrote, we have no chance of grasping results like this unless we see examples -- preferably several -- of domains which are not regular, not normal, and normal but not regular. In this particular example this is easily done, but unfortunately many of the core counterexamples in the subject have a reputation of being too difficult to show beginners. At this point I feel the need to quote directly from p. 136 of Reid's Undergraduate Commutative Algebra:
The catch-phrase "counterexamples due to Akizuki, Nagata, Zariski, etc. are too difficult to treat here" when discussing questions such as Krull dimension and chain conditions for prime ideals, and finiteness of normalisation is a time-honoured tradition in commutative algebra textbooks (comparable to the use of fascist letters $\mathfrak{P}$ and $\mathfrak{m}$ etc., for prime and maximal ideals). This does little to stimulate enthusiasm for the subject, and only discourages the reader in an already obscure literature; I discuss here three counterexamples (taken, with some simplifications, from the famous "unreadable" appendix to [Nagata]) to show some of the ideas involved.
This is very well said (well, except that I honestly don't know what's wrong with $\mathfrak{m}$...): most of the standard texts in commutative algebra leave unanswered the natural questions an alert reader will have: is this hypothesis necessary? is the converse of this result true? What happens if we don't assume that $M$ is a finitely generated module over a Noetherian domain? and so forth.
By a coincidence I have just finished -- that is, within the last half hour -- teaching a first graduate course on commutative algebra. I tried to spend a lot of time on examples, and I was not afraid to make "technical" digressions about what happens when $M$ is not a finitely generated....Especially I spent an extra long amount of time on module-theoretic questions, which made me feel closer to the heart of the subject. It is easy to motivate the need for modules to be finitely generated: there is a structure theorem for finitely generated modules over a PID but there is no structure theorem for infinitely generated abelian groups. The example of $\mathbb{Q}_p$ as a $\mathbb{Z}_p$-module shows that even over a DVR infinitely generated modules can have a complicated structure. Then, when I got to Noetherian rings I motivated them in part by showing that the Noetherian condition was equivalent to many seemingly innocuous and desirable properties, like every submodule of a finitely generated module being finitely generated. At the same time I discussed plenty of examples of non-Noetherian rings, including rings which are very nice "except that they are non-Noetherian" like the ring of all algebraic integers. So I think I gave my students at least an opportunity to feel their way around finiteness conditions in the subject.
Let me add that there are some recent texts which do a much better job at this. Most of all I can enthusiastically recommend T.Y. Lam's Lectures on Modules and Rings. As with all of his books, his skill at balancing theory and examples is superior and makes for very pleasant, stimulating reading.
It goes much the same for compactness in elementary analysis, but it seems easier to me to supply the necessary counterexamples: every time you encounter a theorem which holds on a compact interval $[a,b]$, ask yourself whether it holds on noncompact intervals (and, if applicable, compact non-intervals!). In all the instances I can think of now, such counterexamples are well known and relatively easy to supply.
1) If you are genuinely seriously interested in mentoring undergraduates in REU projects, then you would naturally be fantasizing about projects you might do. If you aren't all that interested in it, you can't fake it just by thinking up some projects. Note that there is some middle ground between 'genuinely seriously interested' and 'not all that interested'. Thinking up projects does show that you are interested in it. The ones you think up don't need to be great; they just need to be not delusional. Keep in mind that weaker schools will only have someone who is potential grad student material only once every several years, so those schools, if they are interested in undergraduate research, will want projects that someone who is not grad student material can tackle. (As a rough and not particularly accurate guide, by 'weaker school' I mean median SAT of incoming freshman below 1150 or so for a small liberal arts college; larger universities may have some stronger students even if the average is somewhat lower.) People in applied and computational areas have a huge advantage here because they can propose research that does not involve proving any theorems.
The truth is that you'll probably have a failure or two before you have a success, and it's not a big deal unless it is a total failure where the student gets nothing out of it. So don't worry too much about the specifics of the project; the point of having one is to show you are serious and not completely delusional (given the school and its students) about it.
2) Unless the advertisement does not ask for a research statement or asks for undergraduate research to be addressed elsewhere, it goes in a section of your research statement.
Keep in mind that many schools that care seriously about undergraduate research will care only a little about your research except inasmuch as it generates projects for undergraduates. When applying for positions at such schools, you may need to reshape your research statement to say only generalities about your own research (since no one there will understand most of it anyway) and focus on the accessible parts and the undergraduate research. It may also be the case that such a school will not ask for or read a research statement, in which case you need to be prepared to put the material as a section in your teaching statement.
3) It absolutely does NOT have to be in your primary research area, as long as it is an area where you have enough of an idea what is going on to actually guide a project. (In other words, you should know enough that having the project turn out to be a fairly well-known solved problem is not at all a risk.) In fact, many professors at undergraduate institutions have gradually migrated out of their dissertation areas to research areas that are more friendly to their students.
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I'm director of graduate admissions for the math Ph.D. program at a research I department. We are certainly interested in assessing research potential. But we don't judge this by published papers; almost no undergrad has any, and when they do, the actual material they contain is usually not of great interest.
It could certainly be helpful to start exploring research. You should do this under the supervision of a professor (whether at your own institution in the context of a thesis or capstone project, or at an REU) who will be able to attest, in a recommendation letter, that you are very likely to prove interesting theorems in the future.