I haven't checked the details but my guess is that Hanna's formula comes from expressing the Genocchi numbers in terms of central factorial numbers (see mathoverflow.net/questions/471918/…) and then using the ordinary generating function for central factorial numbers, which can be found at oeis.org/A036969.

You might look at Richard Stanley's paper Invariants of finite groups and their applications to combinatorics, Bull. Amer. Math. Soc. (N.S.) 1(3): 475-511 (1979), ams.org/journals/bull/1979-01-03/S0273-0979-1979-14597-X

The numbers $\frac12\binom jk \binom{j+k}{k}$ are sequence A092370 in the OEIS (oeis.org/A092370). That's how I found the formula.

For more on plethysm of symmetric functions you might look at Nicholas A. Loehr & Jeffrey B. Remmel, A computational and combinatorial exposé of plethystic calculus. J Algebr Comb (2011) 33: 163–198 DOI 10.1007/s10801-010-0238-4, link.springer.com/article/10.1007/s10801-010-0238-4

You might look at S. Devitt and D. M. Jackson, The enumeration of covers of a finite set, J. London Math. Soc.(2) 25 (1982), 1-6, and the references cited there.

I don't know anything about this sum.

I believe that these numbers are coefficients of the Carlitz-Riordan $q$-Catalan numbers, oeis.org/A227543.