Let $\iota: \mathcal{A}\subset \mathcal{C}$ a replete, full, reflexive subcategory with $F: \mathcal{C}\to \mathcal{A}$ left adjoint to $\iota$, and $\eta_X: X\to F(X)$ the canonical unity.
The canonical counity $\epsilon$ is a isomorphism (because $\iota$ is full and faithfull) and $F(\eta_X)$ is a isomorphism (triangle identity).

Let $W:= \{ f \in |\mathcal{C}|_1\ |\ F(f)\ is\ Iso \}$ (in your post you can consider the saturation...).

Let $\mathcal{B}\subset \mathcal{C}$ the full subcategory of $W$-replete objects defined as in your post.

Claim: $\mathcal{A}= \mathcal{B}$: Let $A\in \mathcal{A}$ and $f: X\to Y$ by $F(f)\in Iso$, considering the square of $f, F(f), \eta_X, \eta_Y$, by universal property follow that $A\in \mathcal{B}$.

Let $B\in \mathcal{B}$, we have to show that $\eta_B\in Iso$, considering $\eta_B\in W$ follow a morphism $g: F(B)\to B$ with $1_B= g\circ \eta_B$, and from
$(\eta_B\circ g)\circ \eta_B=1\circ \eta_B $ follow that (universal property): $\eta_B\circ g=1$.

**Edit**:

COnsidering the case $\mathcal{A}= \mathcal{C}[\mathcal{W}^{-1}]$ for some class of morphisms $\mathcal{W}$, and suppose that we have a full immersion $\iota: \mathcal{C}[\mathcal{W}^{-1}]\to \mathcal{C}$ left adjoint of the natural functor $F: \mathcal{C}\to \mathcal{C}[\mathcal{W}^{-1}]$, and let $W$ defined as above, then $W$ is the saturation of $\mathcal{W}$ and $\mathcal{C}[\mathcal{W}^{-1}]\cong \mathcal{C}[W^{-1}]$.

THen the $W$-local objects are the $\mathcal{W}$-local objects, and this is a replete class.

**Edit** ABout the second question, after the **Zhen Lin** counterexample, I can say only this:
From [P], T.13.11, p.98, given an adjuntion $<\iota, F>: \mathcal{C}\to\mathcal{A}$ with $\iota$ full and faithfull we have that $\mathcal{A}$ is equivalent to $\mathcal{C}[W^{-1}]$ where $W$ is the class of morphisms $f$ such that $F(f)$ is a isomorphism, and $W$ is also the class of morphism's $w$ such that $\mathcal{C}(w, A)$ is a isomorphism (bijection) for any $A\in\mathcal{A}$.
Then if $\mathcal{A}\subset \mathcal{C}$ is the full subcategory of the $\mathcal{W}$-stable objects for some class of morphisms $\mathcal{W}$, then $W$ is the "Galois-closure" of $\mathcal{W}$.

Furthermore, about the existence of the adjoint $\iota$: given a class lef-calculable of morphisms $\mathcal{W}$ of a category $\mathcal{C}$, let $F: \mathcal{C}\to \mathcal{C}[\mathcal{W}^{-1}]$ the natural functor. From [P] 15.3 follow that $F$ has a right adjoint (and it is full and faithfull) IFF for any $X\in \mathcal{C}$ there exist a morphism $u_X: X\to X'$ where $X'$ is $\mathcal{W}$-stable and $u_X$ belong to the saturation $W$ of $\mathcal{W}$ IFF for any $X$ the category $X\downarrow W$ (the full subcategory of $X\downarrow\mathcal{C}$ of the $W$'s with $X$ as domain) has a final object (I seem that the proff work without the left-calculable hypothesis).

Biblio:
[P]: Theory of Categories , Nicolae Popescu & Liliana Popescu