These kinds of questions seem to be treated only in scattered papers over many years. Concerning power maps, see two related papers:
Chatterjee, Pralay, On the surjectivity of the power maps of semisimple algebraic groups. Math. Res. Lett. 10 (2003), no. 5-6, 625–633.
Steinberg, Robert, On power maps in algebraic groups. Math. Res. Lett. 10 (2003), no. 5-6, 621–624.
Both papers investigate when the map is surjective, as do two more recent papers by Chatterjee focusing on local fields. Steinberg's paper works in arbitrary characteristic. In both cases algebraic groups need to be connected.
I suspect there is no comprehensive paper of the kind wanted. The interaction between morphisms and algebraic group structure has been studied over many decades, but not definitively. As Allen remarks, an old theorem of Rosenlicht shows for example that for connected groups an identity-preserving morphism to the multiplicative group is a character. There are several simple modern proofs, one being readily accessible online:
Broughton, S. A., A note on characters of algebraic groups. Proc. Amer. Math. Soc. 89 (1983), no. 1, 39–40.