Let me copy Definition 4.1 of
Libor Barto, Jakub Oprsal, Michael Pinsker
The wonderland of reflections
Israel Journal of Mathematics 223 (2018), 363-398
Defn. 4.1
Let $\mathbf{A}$ be an algebra with signature $\tau$. Let $B$ be a set, and let
$h_1\colon B\to A$ and $h_2\colon A\to B$ be functions. We define a $\tau$-algebra $\mathbf{B}$ with domain $B$ as follows: for every operation $f(x_1,\ldots,x_n)$ of $A$, $\mathbf{B}$ has the operation
$$(x_1,\ldots,x_n) \mapsto h_2(f(h_1(x_1),\ldots, h_1(x_n))).$$
We call $\mathbf{B}$ a reflection of $\mathbf{A}$. If
$h_2 \circ h_1$ is the identity function on $B$, then we say that $\mathbf{B}$ is a retraction of $\mathbf{A}$.
The construction in the problem statement above is the special case
of a reflection where $B=\textrm{im}(f)$, $h_2=f$, and $h_1$ = the inclusion of $B$ into $A$.
In this case, the algebra $\mathbf{B}$ of Definition 4.1
is what would be called $\mathbf{A}_{h_2}$ in the problem statement.
The idempotent construction in the problem statement
is exactly what is meant in Theorem 4.1 by
a retraction. (If $f$ is idempotent in the problem statement,
let $B=\textrm{im}(f)$, $h_2 = f$, $h_1$ = the inclusion of $B$ into $A$
to see that $\mathbf{A}_f$ is what is called a retraction of $\mathbf{A}$
in the wonderland paper.
Conversely, if $\mathbf{B}$ is a retraction of $\mathbf{A}$
in the sense of Definition 4.1, let $f = h_1\circ h_2$
in the problem statement and notice that
$h_1\colon \mathbf{B}\to \mathbf{A}_f$ is an isomorphism.)
Now let me summarize some definitions from the wonderland paper.
A term has height 0 if it is a variable and
has height 1 if it is a fundamental operation
applied to variables. A term has height at most 1
if its height is 0 or 1. An identity $s\approx t$
has height 1 if both $s$ and $t$ have height 1, and
has height at most 1 if both $s$ and $t$ have height at most 1.
The wonderland result that is relevant to this problem is:
Corollary 5.4:
Let $K$ be a nonempty class of algebras of the same signature $\tau$.
(i) $K$ is closed under reflections and products if and only if
$K$ is the class of models of some
set of $\tau$-identities of height 1.
(ii) $K$ is closed under retractions and products if and only if
$K$ is the class of models of
some set of $\tau$-identities of height at most 1.
With this background, let me address the questions in the problem.
Is there a $\mathbb{K}$ such that $F_i(\mathbb{K})\subsetneq F(\mathbb{K})$?
Yes, let $\mathbb{K}$ be the class of semilattices.
It can be shown that the associative law is not a consequence
of height 1 identities, so
$\mathbb{K}\subsetneq F_i(\mathbb{K})$. Also, the idempotent law $x\wedge x\approx x$
has height at most 1, so it will be satisfied in
$F_i(\mathbb{K})$. (This fact is easy to prove from the definitions, you don't need to cite the wonderland paper.)
The class $F(\mathbb{K})$ does not satisfy
$x\wedge x\approx x$, because this class contains nontrivial
algebras of size $>1$ that have constant multiplication.
To see this, choose a semilattice $\mathbf{A}$ and a nonconstant function
$f\colon A\to A$ such that $\textrm{im}(f)$ is a proper subsemilattice
of $\mathbf{A}$
and $f\circ f$ is constant. In this situation $\mathbf{A}_f$
will have size $>1$ and have constant multiplication, so it will
not satisfy the idempotent law, so it will lie in
$F(\mathbb{K})-F_i(\mathbb{K})$.
Is the equational theory of $F_i(\mathbb{K})$ the (deductive closure of the) set of basic equations true in each element of $\mathbb{K}$?
Yes, as indicated by Corollary 5.4 of the wonderland paper.
Related remarks.
Remark 1. The phrase 'basic equations' goes back to
Kelly, David
Basic equations: word problems and Mal’cev conditions.
Abstract 701-08-04, AMS Notices 20 (1972) A-54.
For Kelly, a term is basic if it is a variable, a constant,
or a fundamental operation symbol applied to variables
and constants. An identity is a basic equation if both sides
are basic. This is close to but different from what is meant
in the problem on this page.
Remark 2. The part of Corollary 5.4 that concerns
retractions was observed earlier in
Taylor, Walter
Simple equations on real intervals.
Algebra Universalis 61 (2009), no. 2, 213-226.
I summarize Taylor's Theorem 2.1 as follows:
Theorem 2.1. Let $V$ be a variety
Then $V$ is closed under
set-retractions iff $V$ is definable by simple equations.
Taylor's terminology is a little different than that of the wonderland paper,
but the result is the same in the case of retractions.
(Simple = height at most 1.)
You can find a extended discussion of Theorem 2.1
in Volume 2, Section 6.7, of the encyclopedia
Ralph S. Freese,
Ralph N. McKenzie,
George F. McNulty,
Walter F. Taylor
Algebras, Lattices, Varieties
Mathematical
Surveys
and
Monographs, American Mathematical Society,
v. 268, 2022.
[In particular, FMMT show in their Theorem 6.61
that if $\mathbb{K}$ is the class of semilattices,
then $\mathbb{K}\subsetneq F_i(\mathbb{K})$,
because the latter does not satisfy the associative law.]
Remark 3. The wonderland paper was written primarily to develop tools
to study the Constraint Satisfaction Problem for $\aleph_0$-categorical structures.
