Some premise seem me needed. I’m focused in econometrics since several years ago and pretty earl I realized that the meaning and scope of econometric models was frequently misunderstood and in part can be debatable. Infact definitions and point of views of Professors and researchers was not always the same. Today I’m convinced that the main scope of econometrics are two: causation and prediction; and them must not be conflated.
I'm starting to study econometrics from the Wooldrige's book. But some doubts arise regarding the structural model, its definition and its comprehension in general
If you are focused on structural models the first thing that you have to note is that them have, or should be, causal meaning and causal scope. These discussion can help:
Difference Between Simultaneous Equation Model and Structural Equation Model
Structural Equation Model and Causality in Economics
Unfortunately, the treatment of causality in econometrics textbook are frequently ambiguous and sometimes contradictory. I feared in the past that something strange there was but this article: Regression and Causation: A Critical Examination of Six Econometrics Textbooks - Chen and Pearl (2013); revealed me that the situation is worse. This is a big problem for econometrics today yet. These discussion is related:
Regression and causality in econometrics
linear causal model
I analyzed a dozen of econometric books and I think that: Mostly Harmless Econometrics - Angrist and Pischke (2009) and Econometric Analysis of Cross Section and Panel Data – Wooldrige (2010), the book you cited, are the best among them about causality. The first is probably the best even if it use, let me say, the "experimental paradigm" and not the "structural" one. However these two book, not cited in Chen and Pearl (2013), even if do not encounter some problems, do not follow all the suggestion and tools indicated in the critical article. Tools developed in statistics for causal inference and not in econometrics literature. It seems me that, at least in Pearl opinion, an econometric book that properly address causal questions is not written yet.
So, about your questions 1 and 3 considered together.
Basically, correlation between two random variable like $X$ and $Y$ can come from any source. In its proper meaning correlations, and any statistical association in general, ignore the “rest of the world”, where the data come from and how are collected. At the other side causal relations want to say something about the data generating process, that is treated precisely as a structural equation; in it only the "right data" in the "right position" should be considered
As example we can consider this structural equation:
$Y = \beta_0 + \beta_1 X + u$
Here is important to realize that $=$ is not only the usual algebraic sign but it imply “assignment”, like in computer programming code. Probably this is the main difference between an equation in general and the structural one. This equation imply that $X$ cause $Y$ and no other way around. The structural equations want to formalize “laws of nature” and them come from a more or less developed theory (even just some idea).
Algebraically is always possible to write
$\beta_1 X = Y - \beta_0 - u$
but this equation have no structural meaning.
From the structural equation given we have that $\rho(X,Y) \neq 0$ ($\beta_1 \neq 0$). Informally you can think about that as a sort of "direct correlation". Here, taking given the “mantra” correlation do not imply causation seems me useful to consider that, taking apart special theoretical case, causation imply correlation (read here: Does causation imply correlation?).
$\beta_1$ produce correlation also but, ideally, it come from intervention, like in experiments, and not passive observation, like general statistical associations deal with.
For concreteness you can think at: $Y$ like consumption, $X$ income, $\beta_1$ marginal propensity to consume.
About your point 2 I suppose that the Author intend to speak about the exogeneity condition, like $E[u|X]=0$.
It is a key assumption in econometrics models. Several ambiguity and confusions move around that assumption but them come to solve if you realize that, at least in econometrics, it must be referred on an structural error and not an regression error. These discussions can help:
What is the actual definition of endogeneity?
Zero conditional expectation of error in OLS regression
Endogeneity testing using correlation test
Does homoscedasticity imply that the regressor variables and the errors are uncorrelated?