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Although most model structures appearing "in practice" do not have a reflective subcategory of fibrant objects (unless every object is fibrant), the following result of A. Salch provides many examples that do. Theorem. Let $\mathcal{C}$ be a complete and cocomplete category, and let $\mathcal{A}$ be a full subcategory which is reflective and replete (i.e., ...
The 1-categorical question is well-defined, and its answer is no, in general. I will provide examples below. Lemma 1. Let $\eta \colon X \to F$ be a map from $X$ to a fibrant object, initial among such maps. Then $\eta$ is a trivial cofibration. In particular, $F$ is a fibrant replacement of $X$. Proof. $\require{AMScd}$ Factor $\eta$ as $\eta = pi$, where ...