There has been a good deal of impressive **lattice-theoretical** work on 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.

The focus is on the automorphism group, its orbits, and first order definability.
See for instance

>Cholak, Peter A.; Downey, Rodney; Harrington, Leo A. On the orbits of computably enumerable sets. J. Amer. Math. Soc. 21 (2008), no. 4, 1105–1135.

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.