For any $k$ sums $T_k = 1/|G|\sum_{g\in G} g^k$ belong to the center of the group algebra, for finite group G. For $k=2$ they "are" (up to details and interpretation) Frobenius-Schur indicators. For $k>2$ similar things called "higher" FS indicators.

Question 1 Any element of the center of C[G], and $T_k$ in particular, acts by scalars in any irreducible representation. For $k=2$ the scalar can be $-1,0,1$ and brings important group theory information on the type of irrep (see WP and/or this MO question).

Is it true that values of higher $T_k$ in irreps are also integers ? Do they have some group theory interpretation ? (These MO questions 1, 2 discuss FS and relate them to dim of invariants on tensor powers of representation, but it seems different from my question.)

Question 2

How far is the linear space/ algebra generated by $T_k$ from the whole center of group algebra? ( What can be said in general and in particular for S_n,A_n, GL_n(F_q), UT(n,q) )?

PS

Bonus Question 3 WP- article contains the following remark:

It resembles the Casimir invariants for Lie algebra irreducible representations.

In the sense that Casimirs are the center of U(g), while T_k are in the center of C[G].

Question: are there some more precise analogies ?

In some my research we studied the center of U(gl_n) and its loop algebra,
so such expression resembles "Gelfand" generators of U(gl_n) (which are traces of matrices "E^k", "E" given here), but I do not see any way to go beyond "resemble". While it would be nice seems now I am trying to think on some finite group questions.