Category with no empty hom-sets from a given category. Is there any kind of "completion" to obtain a category for which there are no empty hom-sets from a given category?
 A: The question statement needs to be cleaned up. This question has nothing to do with preorders.
Let $Cat$ denote the category of categories and let $Cat'$ be the full subcategory of categories which have at least one arrow in each homset. The most straightforward reading of this question is 


Does the inclusion $Cat' \to Cat$ have a left adjoint?


One expects the answer to be no, because the definition of $Cat'$ is rather unnatural. In fact, the answer is no, because the inclusion does not preserve limits. To show this, it suffices to specify two arrows in $Cat'$ whose equalizer (in $Cat$) does not lie in $Cat'$.
Well, let's try the simplest possible thing. Let $C$ be the category with two objects $X,Y$, and morphisms freely generated by $f: X \to Y$ and $g: Y \to X$. Let $D$ be the category with two objects $X,Y$ and morphisms freely generated by $f: X \to Y $ and $g_1,g_2: Y \to X$. For $i=1,2$, let $F_i: C\to D$ be the unique functor sending $X \mapsto X$, $Y \mapsto Y$, $f \mapsto f$, and $g \mapsto g_i$. Then the equalizer of $F_1$ and $F_2$ is the arrow category (objects $X,Y$ and a single non-identity arrow $f: X \to Y$), which does not lie in $Cat'$.
On the other hand, let $Cat_*$ denote the category of categories enriched in pointed sets. So an object is a category equipped with arrows $0_{xy} \in Hom(x,y)$ for each homset, such that $0_{xy} f = 0_{xz}$ for every $f: y \to z$ and $g0_{xy} = 0_{wy}$ for every $g: w \to x$. A morphism is a functor $F$ such that $F0_{xy} = 0_{FxFy}$. Then the obvious functor $Cat_* \to Cat$ has a left adjoint, induced by the left adjoint to the natural functor from pointed sets to sets. So if $C$ is a category, its image under the reflection $Cat \to Cat_*$ is the category $\tilde C$ where $Hom_{\tilde C}(x,y) = Hom_C(x,y) \amalg \{0_{xy}\}$ and composition defined in a way that you can hopefully guess.
We can also consider a category whose objects are similar to the objects of $Cat_*$ as described above, except without the conditions that $0_{xy}f =0_{xz}$ and $g0_{xy} = 0_{wy}$ (and morphisms as above). There is also be a left adjoint to the inclusion of this category into $Cat$: it sends $C$ to the category freely generated by $C$ along with one new morphism in each homset (by taking composites, though, we end up with lots of new morphisms in each homset).
A: There is a canonical construction along the lines you request:
$O: Cat\rightarrow{Set}$ 
the functor assigning to each small category the set of its objects, has a right adjoint
$M: Set\rightarrow{Cat}$ the functor which assigns to each set $X$ a category with objects $X$ and exactly one arrow in each hom set.
Now, think about it: in a category where you have exactly one arrow between any objects $a\rightarrow b$ and exactly one $b\rightarrow a$ and exactly one $a\rightarrow  a$ and one $b\rightarrow  b$ (the identities),  $a\rightarrow b$ and $b\rightarrow a$ must be inverse of each other, that is $a\rightarrow b$ is an isomorphism.  
Moral: all objects in the categories obtained by applying $M$ are isomorphic to each other
That's why I called this functor $M$, because it "merges" all objects in the original category into an isomorphic blob.
I am not aware of an official name for functor $M$. Please let me know if there is one?
Mac Lane's CWM 2nd edition page 90  ex. 9 describes this construction.
