There has been a good deal of impressive **lattice-theoretical** work on automorphisms and definability in the lattice of **recursively enumerable sets**. (These are also known as r.e. sets, computably enumerable sets, and c.e. sets.)

This work in particular has dealt with the lattice of c.e. subsets of a given c.e. set $E$, $\mathcal L(E)$, which has different properties depending on properties of $E$ such as hypersimplicity and creativity.

See for instance Peter Cholak, Automorphisms of the lattice of recursively enumerable sets, Memoirs of the AMS, 1995.

This topic is a subtopic of recursively enumerable sets and degrees (AMS classification 03D25).

But, I'm not aware of your bullet point (2) regarding functions with domain $E$ being incorporated in this study.