Let $C$ be a category and $D$ be a full subcategory such that every object of $C$ has a monomorphism to some object of $D$ (or dually, an epimorphism from some object of $D$).
Is it then true that any epimorphism (resp. monomorphism) in $D$ must in fact be an epimorphism (resp. a monomorphism) in $C$?
If so, then this would imply that any epimorphism in a category $D$ with finite limits is also an epimorphism in the reg/lex completion of $D$ (though it will not be regular there unless it was a split epimorphism to begin with). In particular, if $D$ is any finitely complete category with a non-split epimorphism, then $D_{reg/lex}$ will not be balanced, as the mono part of the regular epi-mono image factorization in $D_{reg/lex}$ of any non-split epimorphism in $D$ will be a morphism that is both monic and epic but not an isomorphism.
Sure. Let $f:x\to y$ be an epimorphism in $D$ and let $g,h:y\to z$ be two morphisms in $C$ with $gf=hf$. We have to show $g=h$. Let $k:z\to w$ be a monomorphism in $C$ with $w\in D$. Then $kgf=khf$. Since $f$ is an epi in $D$ and $kg$ and $kh$ are morphisms of $D$ ($D$ being full), we have $kg=kh$. Since $k$ is a mono in $C$, we have $g=h$ as desired.