Does universal construction apply to any arbitrary category? Does universal construction presuppose a category equipped with morphisms that implement an order? 
Otherwise how does the universal construction pick the most suitable element for the given purpose? 
How does "more suitable" or "best" is defined for an arbitrary category?
 A: There's nothing like "best" or "most suitable" around a universal property, it's always uniqueness of certain arrows/diagrams, even in a non-thin category.
Moreover, everything with universal property is simply an initial object 'somewhere', i.e. in a suitable category $\mathcal B$, it is an object $O$ such that for every object $B$ there is one and only one morphism $O\to B$.
For instance, its dual, a terminal object of a category $\mathcal A$ is an initial object of the opposite category $\mathcal A^{op}$. 
The coproduct of objects $A_i$ is an initialal object in the category of 'cocones over $A_i$'s'
[ which has the cocones $(\underset{A_i\to X}{f_i})_i$ as objects and those  $u:X\to Y$ morphisms as morphisms $(f_i)_i\to (g_i)_i$ which satisfy $u\circ f_i = g_i$ for each $i$.] 
Similarly, any colimit can be seen as an initial object of the category of cocones over the given diagram. 
Dually, any limit is a terminal object of the category of cones over the given diagram, so that it's an initial object of its opposite category.
