1. Is there a simple explanation of how they estimated the "teacher quality" variable? The paper is written in a very complicated and abstruse way, and I don't have time to wade through it, but surely the basic idea, if valid, should be explicable in a paragraph of plain English.

One of the main points of the paper is that typical measures (teacher experience, education, etc.) are not good predictors of quality. The authors spend a lot of time developing a phenomenological model of teacher quality based on comparing student achievement within a school, to reduce the impact of greater variation in populations between schools. From the paper:

3. THE IDENTIFICATION OF TEACHER EFFECTS In this section we develop an estimator of the variance of teacher quality that avoids problems of student selection and administrator discretion that potentially have biased prior attempts. This estimator is based upon patterns of within-school differences in achievement gains and ignores variations in teacher quality across schools, because such variation cannot readily be disentangled from student differences and the contributions of other school factors. This strategy yields a lower bound estimator for the importance of teacher quality that relies upon minimal maintained assumptions about the underlying achievement process. Importantly, we do not focus solely on measurable characteristics of teachers or schools as is typically done in this literature but instead rely on student outcomes to assess the magnitude of total teacher effects, regardless of our ability to identify andmeasure any specific components. This semiparametric approach provides both an estimate of the role of teacher quality in the determination of academic achievement and information on the degree to which specific factors often used in determining compensation and hiring explain differences in teacher effectiveness.

2. Even if we take the findings of the paper at face value, the "$100 trillion" estimate is a complete non sequitur. Can the entire effect really be purely because better teachers impart greater wealth-producing skills? Or could it be, at least partly, because they impart advantages in zero-sum signaling and rent-seeking games?

This claim is developed in a different paper (linked in the first link I posted), which draws from the paper I linked to discuss teacher quality. Unfortunately, that paper is paywalled, but I have extracted the relevant part, section 4.2, as a pdf (only 2 pages).

They used data comparing performance on math and science tests to economic growth for different countries. They then calculated the improvement in economic growth due to an improvement in student performance from replacing lower-quality teachers with higher-quality teachers. Obviously there is a simplifying assumption of linearity being made for the correlation of test performance and economic growth, and test performance as a measure can fall afoul of Campbell's Law.