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.

Here is a somewhat positive answer in terms of ###germs and equivalence modulo finite differences. ###

There has been a good deal of impressive lattice-theoretical work on the lattice of recursively enumerable sets. (This topic is a subtopic of recursively enumerable sets and degrees (AMS classification 03D25); 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.

About half of the work concerns the quotient structures modulo finite differences. So in a sense they are studying germs and there is some analogy with what you are asking for. That is, one studies $\mathcal E$, the lattice of r.e. sets, and $\mathcal E^*$, the lattice of equivalence classes of elements of $\mathcal E$, where $A$ and $B$ are equivalent if $$ \{n : A(n)\ne B(n)\} $$ is a finite subset of $\omega=\mathbb N$.

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.

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.

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.