Short answer

• (a) is classical.
• (c)–(d) are standard “scaled” variants of (a) and can be found in tables.
• (e)–(h) follow from Jonquière’s inversion formula for polylogarithms together with the Mellin–Beta evaluation used in (a) and the classical integrals ∫ ln^m(x)/(1+x^2). They are not usually tabulated in exactly your binomial/Euler-number form, but they are direct consequences of well-known general identities.
• (i)–(j) are depth‑1 alternating Euler sums with odd denominators (level 4). Their reducibility to linear combinations of zeta-, eta- and beta-values is well documented; your particular closed forms follow from the standard generating‑function method.

More details and references

1. The base integral and its scaled variants

• The identity ∫0^∞ ln^{2n}(x)/(1+x^2) dx = |E{2n}| (π/2)^{2n+1} is classical; it follows from ∫_0^∞ x^{s-1}/(1+x^2) dx = (π/2) csc(π s/2) and the relation between derivatives at s=1 and Euler numbers via csc(π s/2) = sec(π(1−s)/2). References: • NIST DLMF, §5.4 (Beta integrals) and §24.2 (Euler numbers and secant/tangent expansions).
  • A. P. Prudnikov, Yu. A. Brychkov, O. I. Marichev, Integrals and Series, Vol. 1 (and/or Vol. 2), entries on ∫ ln^m x/(x^2+a^2).
  • Gradshteyn & Ryzhik, Table of Integrals, Series, and Products (8th ed.), formulas in 4.295 cover ∫_0^∞ ln^m x/(x^2+a^2) dx; setting a=1 yields (a), and the general a (or your y) gives (c)–(d) after the substitution x↦x/√y.

  Your formulas (c) and (d) are exactly the binomial expansion obtained from ∫0^∞ ln^{m}(x)/(x^2+a^2) dx = (π/2a) times a polynomial in ln a with coefficients built from |E{2k}|; see GR §4.295 (even m) and the vanishing of the odd moments at a=1.

2. The polylogarithmic integrals

• The key identity you use, Li_a(−z) + (−1)^a Li_a(−1/z) = −2 ∑_{k=0}^{⌊a/2⌋} η(2k)/(a−2k)! ln^{a−2k}(z), is a specialization of Jonquière’s inversion formula for polylogarithms (with argument on the negative ray). References: • L. Lewin, Polylogarithms and Associated Functions, North‑Holland, 1981; see Chapter 1 (Jonquière’s inversion) and Chapter 7 for integral representations.
  • NIST DLMF §25.12(ii) (polylogarithm functional equations).

• Inserting the standard integral representation of Li_s and evaluating the inner rational integral with (a)–(d) yields (e)–(h). This is a routine application of Mellin transforms and the Beta function: ∫_0^∞ x^{s−1}/(1+x^2) dx = (π/2) csc(π s/2), together with ∫_0^1 ln^m x/(1+x^2) dx = (−1)^m m! β(m+1), and ∫_0^1 ln^m x/(1−x) dx = (−1)^m m! ζ(m+1), ∫_0^1 ln^m x/(√x(1−x)) dx = (−1)^m m!(2^{m+1}−1) ζ(m+1). References for these integrals: • Gradshteyn & Ryzhik, §4.291–§4.293, §4.384 (the β integral).
  • P. Flajolet and B. Salvy, Euler sums and contour integral representations, Experimental Mathematics 7 (1998), 15–35, which develops exactly this Mellin–polylog machinery and gives general reductions to η-, ζ-, and β-values for integrals of Li’s against rational kernels such as 1/(1+x^2).

  Your explicit binomial/Euler-number coefficient form is a clean packaging of what one obtains by differentiating with respect to a Mellin parameter and then expanding via Jonquière; while I am not aware of the exact equations (e)–(h) in that literal form in a handbook, they follow directly from these standard tools and are thus essentially known.

3. The harmonic sums (Euler sums)

• Using ∑_{k≥1} H_k^{(m)} x^k = Li_m(x)/(1−x) and the integral ∫_0^1 x^k ln^n x dx = (−1)^n n!/(k+1)^{n+1} is the standard generating‑function route to Euler sums. Your identities (i)–(j) are depth‑1 alternating Euler sums with odd denominators (level 4). It is well known that such sums reduce to rational linear combinations of products of ζ(odd), η(even)= (1−2^{1−2r})ζ(2r), and β-values. References: • D. Borwein, J. M. Borwein, and R. Girgensohn, Explicit evaluation of Euler sums, Proc. Edinburgh Math. Soc. 38 (1995), 277–294.
  • J. M. Borwein, D. M. Bradley, D. J. Broadhurst, P. Lisoněk, Special values of multiple polylogarithms, Trans. Amer. Math. Soc. 353 (2001), 907–941 (see especially §§3–5 on level‑4 sums).
  • D. M. Bradley, Multiple zeta values, Euler sums and their generalizations, in Zeta Functions, Topology and Quantum Physics, Springer (2005), 75–90.
  • K. Hessami Pilehrood and T. Hessami Pilehrood, Simultaneous generation of identities for zeta values and Euler sums, J. Number Theory 131 (2011), 1385–1407.

  Your formulas (i)–(j) are immediate corollaries of (g)–(h) and the above generating function; while the exact binomial/Euler-number presentation may be new in form, the resulting evaluations are encompassed by the literature on Euler sums at level 4.

Summary

• Items (c)–(d): appear in standard tables (scaled moments of the Cauchy kernel; Euler numbers).
• Items (e)–(h): consequences of Jonquière’s inversion and Mellin–Beta evaluations; these reductions to ζ/η/β are known, even if the precise binomial/Euler-number packaging is not commonly tabulated.
• Items (i)–(j): special cases of well‑studied alternating Euler sums (level 4) and known to reduce to ζ/η/β.

So, from the standpoint of mathematical content, all of (c)–(j) are essentially known results; your derivations are correct and align with the established framework.