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.