What limits/colimits are preserved by the Yoneda embedding? I know that the contravariant Yoneda embedding $X\mapsto \mathcal{C}(-,X)$ preserves all small limits that exist in $\mathcal{C}$. I guess it follows that the covariant Yoneda embedding preserves all colimits. 
But what happens if we have a case in which a limit (rather than a colimit) appears in the first argument of hom, i.e we have a functor of the form $\mathcal{C}(\varprojlim X_i,-)$? None of the two cases above apply. Are there cases in which that functor becomes $\varprojlim \mathcal{C}(X_i,-)$ or $\varinjlim \mathcal{C}(X_i,-)$?
 A: The $\text{Hom}$ functor preserves limits in each argument (in a very strong sense), neither preserves colimits in general.  You should prove this as it's the source of continuity for most other things that are continuous, most notably adjoints. (Because (co)limits in functor categories are computed point-wise, this lifts to the Yoneda embeddings.) 
But be a bit careful, $\text{Hom}(-,X)$ is a functor $\mathcal{C}^{op} \to \mathbf{Set}$, so being continuous means it takes colimits in $\mathcal{C}$ to limits in $\mathbf{Set}$.  (Of course, this is equivalent to taking limits in $\mathcal{C}^{op}$ to limits in $\mathbf{Set}$, i.e. continuity.)
Considering a slightly different problem, if we fix an $X$ and ask what does it mean for $\text{Hom}(X,-)$ to preserve all colimits, you get the notion of a tiny object assuming $\mathcal{C}$ has all colimits (this situation is the same as the one you were talking about, just in the opposite category).  An object being tiny is an unusual thing, for example, in $\mathbf{Set}$, the only tiny object is the terminal object.  It would be unusual for a cocomplete category to have nothing but tiny objects.  In fact, considering different classes of colimits gives different notions of "smallness".  For example, if $\text{Hom}(X,-)$ preserves all filtered colimits, then $X$ is a compact or finitely presented object.  A strongly finitely presented object is one for which that functor preserves all sifted colimits.  A connected object (in an extensive category) is one where that functor preserves coproducts. A duality relative to a limit doctrine gives some general definitions and results for situations like this.
