To me this question is outside the realm of any standard econometric, textbook answer or solution. I can see several approaches to addressing it but no one "correct" or "best" solution.
Personally, I like the panel data model with OLS estimation structure. It makes sense especially wrt pooling the relatively sparse information available for female CEOs. Not to mention that this approach has a long and venerable history in econometric modeling of corporate performance. Just give consideration to transformations to the dependent variable(s) to ensure that it's scale invariant, as appropriate. A key question is whether you use an ANOVA or mixed model (hierarchical) functional form. The latter approach is motivated by the fact that firms can be nested within SIC codes, forming a hierarchical structure. It's been demonstrated that this class of models reduces the variance considerably vs non-hierarchical ANOVA.
You haven't stated what your performance metrics are. This seems like a useful piece of information to know. Lots of metrics suggest themselves such as stock price (or its natural log), year-over-year % return, market cap, total return to shareholders (aka EVA or economic value added), etc. I would have a hard time picking just one. In fact, my view is why would you want to pick just one when MANOVA-type frameworks are available and are potentially quite insightful? I would want to leverage a combination of dependent variables including relative (% return, TRS, etc.) with absolute (stock price, market cap, etc.) measures to assess how the drivers of performance evolve as a function of the type of measure under analysis.
A Bayesian such as Andrew Gelman, in his book with Jennifer Hill Data Analysis Using Regression and Multilevel/Hierarchical Models (chap. 13), would contend that a sample size of 1 remains analyzable, at least in terms of the posterior information produced by Bayesian models. Adopting this framework might make the sparsity of female CEO information more tractable.
In terms of one of the tests you want to perform:
In the second part of my analysis, I am trying to correct for
potential endogeneity problems, and try to see if there is a reverse
causality, meaning do female CEOs really improve the performance of
the firm, OR do they select themselves into better performing firms or
better performing firms have more flexibility to nominate a woman as
CEO.
One immediate problem with this hypothesis as stated is that you haven't indicated how you are measuring firm "flexibility." In other words, it's a great question but how is "flexibility" operationalized? Comparative firm performance is a more readily testable hypothesis by focusing on performance before and after a female CEO is introduced.
Endogeneity is a particularly tough issue to address and even the most hardcore econometric modelers will admit that using IVs introduces as many problems as it solves, i.e., the cure can be worse than the disease. Among the recommended "solutions" is 2SLS. Wooldridge, in his book Econometric Analysis of Cross Section and Panel Data devotes several sections to this discussion. My problem with his book is that it's from a theoretician's perspective, i.e., his examples are based on "toy" data where the input matrices are neat, balanced and completely observed, i.e., he doesn't have to deal with missing values, unbalanced matrices wrt time and sample size, censoring or any other irregularities. This is rarely the reality in applied work and is certainly not the case with your information.
Given that endogeneity is mostly a theoretical problem, the analyst is almost forced to use theory to drive a solution. Empirically, this makes it an almost insoluble, indeterminate problem. Not having read anything in the corporate modeling literature that discusses this specific challenge, I would draw an analogy to "age-period-cohort" models where the aspiration is to disentangle separate effects for each factor independent of the other two. These are issues in fields such as geriatric aging, sociology, marketing or political science, requiring quite long time frames to even initialize the models. (Note age refers to the chronological age of a unit, period refers to a era or set of chronological years, e.g., the 60s or 90s and cohort refers to a a group typically based on birth year, e.g., millennials or baby boomers). Indeterminancy is inherent to the model's structure. For instance, statistician and political scientist, Andrew Gelman has written in an unpublished paper, Thoughts on new statistical procedures for age-period-cohort analyses, (here ... http://www.stat.columbia.edu/~gelman/research/unpublished/apc.pdf), "There can be no general solution to the age-period-cohort problem." He then goes on to say that, "Some methods of constraining the possible space of solutions seem more reasonable than others," but offers little beyond intuition and common sense as selection criteria.
In your case, the relative recency of female CEOs further complicates the problem. This is a kind of censoring which potentially argues for addressing the issue as a survival model. Another approach might be to treat it as a regression discontinuity model where companies with at least one female CEO are your treatment group and companies never having a female CEO are the control group. The effect modeled is performance before and after the introduction, by collapsing or averaging the model-predicted time series, after controlling for the other factors of course, creating an eight cell design (pre vs post, treatment vs control, male vs female).
Some interactions suggest themselves. Most stat packages offer a variety of sum-of-squares metrics in ANOVA-type models, such as hierarchical or independent effects, which can be used in determining the relative importance of the variables in the model. Ulrike Groemping has papers and an R module (RELAIMPO) which review the heuristics suggested in the literature for relative variable importance. Her approach in her R module is to leverage the various ANOVA-type effects in a computationally intensive way to tease out average or overall variable effects under differing model specifications. Her stuff is worth a look.
Anyway, these are just a few thoughts. As already noted, a single answer to your questions will not suffice.
/** Later Thoughts **/
You mentioned "reverse causality." To me, that suggests an interest in determining lead-lag structures without an a priori theory about the direction of the causal structure. This paper by Sornette suggests a nonparametric approach to understanding lead-lag structures in which time's arrow is treated agnostically. Non-parametric Determination of Real-Time Lag Structure between Two Time Series: the “Optimal Thermal Causal Path” Method, Sornette and Zhou.
https://arxiv.org/pdf/cond-mat/0408166.pdf
I fully agree that Pearl's tone is arrogant, and his characterisation of "statisticians" is simplistic and monolithic. Also, I don't find his writing particularly clear.
However, I think he has a point.
Causal reasoning was not part of my formal training (MSc): the closest I got to the topic was an elective course in experimental design, i.e. any causality claims required me to physically control the environment. Pearl's book Causality was my first exposure to a refutation of this idea. Obviously I can't speak for all statisticians and curricula, but from my own perspective I subscribe to Pearl's observation that causal reasoning is not a priority in statistics.
It is true that statisticians sometimes control for more variables than is strictly necessary, but this rarely leads to error (at least in my experience).
This is also a belief that I held after graduating with an MSc in statistics in 2010.
However, it is deeply incorrect. When you control for a common effect (called "collider" in the book), you can introduce selection bias. This realization was quite astonishing to me, and really convinced me of the usefulness of representing my causal hypotheses as graphs.
EDIT: I was asked to elaborate on selection bias. This topic is quite subtle, I highly recommend perusing the edX MOOC on Causal Diagrams, a very nice introduction to graphs which has a chapter dedicated to selection bias.
For a toy example, to paraphrase this paper cited in the book: Consider the variables A=attractiveness, B=beauty, C=competence. Suppose that B and C are causally unrelated in the general population (i.e., beauty does not cause competence, competence does not cause beauty, and beauty and competence do not share a common cause). Suppose also that any one of B or C is sufficient for being attractive, i.e. A is a collider. Conditioning on A creates a spurious association between B and C.
A more serious example is the "birth weight paradox", according to which a mother's smoking (S) during pregnancy seems to decrease the mortality (M) of the baby, if the baby is underweight (U). The proposed explanation is that birth defects (D) also cause low birth weight, and also contribute to mortality. The corresponding causal diagram is { S -> U, D -> U, U -> M, S -> M, D -> M } in which U is a collider; conditioning on it introduces the spurious association. The intuition behind this is that if the mother is a smoker, the low birth weight is less likely to be due to a defect.
Best Answer
First let me say that if I were grading your questions, I would give you an excellent grade. These are the introductory questions of the book, so you still do not have all the tools to think through the problems, but you are already showing that you know you need to take causal information into consideration to answer it.
Now as to the answer, notice the question asks you whether you want to look at the aggregate data or the segregated data. It turns out in both cases you want to look at the segregated data.
In question A, the size of the stone affects both the choice of the treatment and the health status. Hence, you need the segregated data to eliminate this bias and determine which treatment is more effective, either conditionally or unconditionally. To know which treatment is unconditionally better, you need the segregated data to obtain the average causal effect weighted by the probability of stone size $P(Y = 1|do(T)) = \sum_{S}P(Y = 1|T, S)P(S) \neq P(Y =1 |T)$, where $Y$ is health status, $T$ treatment choice and $S$ stone size. If you want to know which treatment is conditionally better, then it's clear you also need the segregated table.
If it's strange to think of picking a treatment for yourself without knowing the stone size, maybe it will be easier to understand the analogous question of having to pick only one treatment to a whole population (say for technical/budget reasons you can't pick both). In this case you want to know which one has the largest average treatment effect on the population as a whole.
Question B is a similar problem, difficulty is a confounder so you need the segregated table to know which doctor is better, both conditionally and unconditionally. Your point about sample size is completely valid, in real life you should always consider sample uncertainty --- but notice it doesn't change the fact that you would still need the information of the segregated data.
Regarding your last comment,
It actually touches a deep problem in causal inference, which is the assumption of invariance. Take the case where the doctor started drinking heavily just now. In this case the data before and after that event do not come from the same causal model --- so you would actually need more information and more causal assumptions to make inference in this case.