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 1b (later edit)

One can generalize this consturction as follows consider free associative ring Z < x_1...x_|G| > . Consider S_|G| invariants in this algebra. Any such invariant can be mapped to the center of the group algebra, just substituting "g" instead of x_|g|. (see MO quest).
It will clearly give central element in C[G].

Is it true that there values will be interges in any irrep ? Moreover by analogy with T_k their values should be divisible by |G|.

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) )?

Same question not about T_k, but also about elements discussed in questions 1b above.

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.