We begin with the observation that the signed generalized Stirling polynomials \(P_k(m,x)\), which occur in a generalization of Malmsten's integral, reduce to the falling factorials when \(k=m\). The structure of these generalized Stirling polynomials is then used to obtain recurrence relations, gamma-polygamma formulas for the polynomials \(P_{m-s}(m,x)\), a more transparent proof of a vanishing identity used in earlier closed forms, and a finite approximation to \(\cosh \pi x\) with a corresponding limit formula for \(\pi\). We also observe that these polynomials occur naturally as signed residues of the equal-period Barnes multiple zeta function, namely
\[P_k(m,x)=(-1)^k m!\operatorname*{Res}_{s=m+1-k}\zeta_{m+1}(s,x).\]In addition, we derive the reflection formula \(P_k(m,m+1-x)=(-1)^kP_k(m,x)\) and use these polynomial identities to obtain explicit identities for Stirling cycle numbers. We then turn to finite nested sums built from the hyperbolic-secant integral sequence \(\chi_n\). After the lower bounds are fixed, the nested sums become coefficient-counting problems: the common-lower-bound case gives binomial coefficients, while the staircase case gives Catalan numbers. Combining these counts with the closed forms for the individual \(\chi_j\)'s produces explicit evaluations involving Catalan's constant, zeta values, and polygamma values at one quarter. A Wolfram Language package accompanies the formulas.