Your requirements are quite stringent! As you know well, ANT is a couple of layers removed from "practice". In general, I find that the methods deriving from the development of algebraic number theory eventually lead to incomparably more applications than any of the standard ANT theorems themselves. Just a few examples that quickly spring to mind: Gauss reduction of quadratic forms → shortest lattice vectors → LLL; Dirichlet units → Minkowski geometry of numbers → convex geometry; class groups and unit groups → finitely generated abelian groups → (pick your favorite application of group theory, e.g. in abelian harmonic analysis). I would try to project this deeper idea over the immediate payoff. Also, unsurprisingly, "elementary" number theory presents more immediate applications, e.g. to cryptography and specifically, to primality testing and factorization.
But enough of philosophy! Here a few concrete applications:
Construction of codes and dense lattice packings using multiplicative groups of global fields by Rosenbloom and Tsfasman (Invent. Math. paper or see Tsfasman's survey "Global fields, codes and sphere packings").
Margulis arithmeticity theorem: not only are algebraic integers useful in constructing discrete groups, but after imposing certain conditions (an irreducible lattice in a higher rank semisimple Lie group), all of them arise in this way! These lattices have been used in constructing Ramanujan graphs and superconcentrators and to uniform distribution of points on spheres (admittedly, arithmetic of the field is rather secondary: you can get good constructions starting from a group over Z).
Lind's theorem on realizability of any Perron–Frobenius integer as the leading eigenvalues of a positive integer matrix. Since log λ is the entropy, this does have semi-practical consequences (cf the textbook of Lind and Marcus).
Lucas–Lehmer primality test, Lucas and Fibonacci pseudoprimes, Grantham's Frobenius test. This strides the border between elementary and algebraic number theory, which may actually be an advantage in an undergraduate class!
I'd be curious to know how do they jibe with your goals.
For part 1 of the question, he would most likely have used the Euler-Maclaurin summation formula
$$
\sum_{n=1}^{\infty}\frac{1}{e^{nt} - 1} = \int_{1}^{\infty}\frac{dx}{e^{xt} - 1} +
\frac{1}{2}\frac{1}{e^t - 1} + \int_{1}^{\infty}S(x)\left(\frac{d}{dx}\frac{1}{e^{xt} - 1}\right)dx
$$
with $S(x)$ the sawtooth function. It is easy to obtain the leading term, because it comes from the first integral
$$
\int_{1}^{\infty}\frac{dx}{e^{xt} - 1} = \frac{1}{t}\int_{t}^{\infty}\frac{du}{e^u - 1}
$$
by the change of variable $u = xt$. We have
$$
\int_{t}^{\infty}\frac{du}{e^u - 1} = \int_{t}^{1}\frac{du}{e^u - 1} + \int_{1}^{\infty}\frac{du}{e^u - 1},
$$
and
$$
\frac{1}{e^u - 1} = \frac{1}{u} + \left(\frac{1}{e^u - 1} - \frac{1}{u}\right)
$$
on $0 \leq u \leq 1$, so that
$$
g(t) = \frac{1}{t}\log\left(\frac{1}{t}\right) + O\left(\frac{1}{t}\right).
$$
But to get the second term looks harder, for the integral with the sawtooth function contributes to that term. To go further, one can integrate by parts in that integral, which is the standard approach, or write it as a sum of integrals over the
intervals from $n$ to $n+1$. Also the sawtooth function has a simple Fourier expansion, which may help. I should remark that the integral with the sawtooth function is $O(1/t)$ as one sees when bounding it by passing the absolute value under the integral sign and using $|S(x)| \leq 1/2$. Anyway, I am pretty sure that part 1 is doable with some work.
Part 2 looks trickier. The Lambert series expansion
$$
\sum_{n=1}^{\infty}(1 + \mu(n))e^{-nt} = \frac{e^{-t}}{1 - e^{-t}} + e^{-t} =
\frac{1}{t} + \frac{1}{2} + O(|t|)
$$
is a little nicer than the one for the divisor function; not only are the coefficients nonnegative, but they are also bounded. Supposing that we have a Tauberian theorem strong enough to yield
$$
\sum_{n \leq x}(1 + \mu(n)) \sim x,
$$
we would then have proved the Prime Number Theorem from the Lambert series. It seems a little unlikely that Dirichlet had such a strong Tauberian theorem; would he not have proved the Prime Number Theorem if he had? Of course, this argument by analogy is not conclusive, since the two
situations differ by a factor of $\log(x)$.
We shall never know what argument Dirichlet had, and he may have found an approach that
did not use a Tauberian theorem, perhaps exploiting special properties of the divisor function. It is worth noting that Voronoi's first proof of the error term $O(x^{1/3}\log(x))$ for the divisor problem was based on the Euler-Maclaurin summation formula.
Best Answer
There are two main sources of repeated logs. (These sources can be further refined into natural subcategories, but I'll only mention a couple of those subcategories.) Those two main sources are:
Type 1: Repeated logs occur because that is just the truth of the matter.
One of my favorite examples is a 2008 theorem of Kevin Ford, solving the multiplication table problem. The theorem states that $$ |\{a\cdot b\, : a,b\in \{1,2,\ldots,N\}\}|\asymp \frac{N^2}{\log(N)^c(\log\log(N))^{3/2}}, $$ where $c=1-\frac{1+\log\log(2)}{\log(2)}$.
Lest you believe that the $(\log\log(N))^{3/2}$ factor is a consequence of this being a 2-dimensional problem, it also shows up in the other dimensions. See this other question for more information.
In some cases it is much easier to see where these extra log's come from. For instance, when turning sums over integers into sums over primes, this often leads to an extra log coming into force, just from the nature of the problem at hand and asymptotics with primes.
For instance, we have $$ \sum_{n=1}^{N}\frac{1}{n}=\log(N)+\gamma+o(1)\ \text{ while }\ \sum_{p\leq N,\ p\text{ prime}}\frac{1}{p}=\log\log(N)+B+o(1), $$ where $\gamma$ and $B$ are well-known constants. These two asymptotics can be thought of as discrete version of the integral equalities $$ \int\frac{1}{x}\, dx = \log(x) \ \text{ while }\ \int\frac{1}{x\log(x)}\, dx=\log\log(x). $$
Since primes occur all over number theory, and they also come weighted with an extra log factor, this often contributes extra double-log factors.
Type 2: Repeated logs occur as an artifact of our current best machinery.
For example, Rankin showed in 1938 that the largest prime gap below $N$, for $N\gg 0$, is at least $$ \frac{1}{3}\frac{\log(N)\log\log(N)\log\log\log\log(N)}{(\log\log\log(N))^2}. $$ These extra logs happen when optimizing inequalities, and when using the known machinery of the day. But they do not represent a fundamental truth about the problem. The constant $\frac{1}{3}$ has been slowly improved. Recently, in 2014, Ford, Green, Konyagin, and Tao improved this bound, and Maynard also did so independently, by replacing the fraction $\frac{1}{3}$ by an arbitrary number. See this preprint and this other preprint for more details. Later, these five mathematicians together removed the square from the denominator.
If you read the proofs, the logs are coming from the current state of the art sieve methods, together with bounding techniques. When you solve for the best fit functions to undo some of the exponentiation that occurs in calculations, the logs just fall out.
In these types of problems, it is not inconceivable (and actually occurs quite regularly) that one new idea is applied to the problem, and the asymptotic changes (sometimes involving more multi-log factors, to account for the small additional room for improvement that was gained). What is surprising about Rankin's bound is that even though it is far from the predicted asymptotic, each extra idea only changed the constant out front---at least until recently.
Edited to add: Working through a well-written proof will, of course, give a deeper understanding of where iterated logs arise in the problem at hand. That is certainly true for the three examples above.
However, if you are not yet familiar with sieve theory, or the circle method, I wouldn't recommend working through the big proofs of those theorems mentioned above and in your question (at least, not initially). Rather, I would recommend starting with an introductory text on sieves, such as Cojocaru and Murty's book "An Introduction to Sieve Methods and their Applications". Double-logs occur almost at the very beginning. Triple-logs show up in the exercises in Chapter 5 (and perhaps earlier). Indeed, problem 25 is a typical example of how a triple log is introduced to improve an asymptotic.