About the Yoneda objects of a locally presentable category This question is a follow-up of Extending functors defined on dense subcategories.

Let $\mathcal{K}$ be a locally presentable category. An object $X$ of
  $\mathcal{K}$ is called a Yoneda object if the functor
  $\mathcal{K}(X,-):\mathcal{K} \to \text{Set}$ is colimit-preserving.

For example, the representable functors $\mathcal{A}(a,-)$ of $\text{Set}^\mathcal{A}$, where $\mathcal{A}$ is a small category are Yoneda objects since $\text{Set}^\mathcal{A}(\mathcal{A}(a,-),F)=F(a)$. A Yoneda object is finitely presentable.

Question 1: Are there other Yoneda objects in $\text{Set}^\mathcal{A}$
  than the representable functors (up to isomorphisms of functors) ?

Every locally presentable category is a reflective subcategory of a category of the form $\text{Set}^\mathcal{A}$.

Question 2: Let $\mathcal{K}$ be a locally presentable category having
  a strong generator of Yoneda objects. Is this category equivalent to a
  category of the form $\text{Set}^\mathcal{A}$ ?

 A: The usual name for "Yoneda objects" is "tiny" or "small-projective".


*

*In general the tiny objects in a presheaf category are the retracts of representables.  In particular, if $A$ is Cauchy-complete, then every tiny object is representable.

*Yes.  In fact, instead of local presentability it is enough to assume cocompleteness; see Theorem 5.26 of Basic Concepts of Enriched Category Theory.
A: These objects are usually called absolutely presentable. In $Set^\mathcal A$, they are precisely retracts of representables. Presheaf categories are characterized as cocomplete categories having a strong generator consisting of absolutely presentable objects (M. Bunge 1969).
A: Question 1: The Yoneda objects are precisely retracts of representables. There are quite a few buzzwords which are relevant here (Cauchy completion, idempotent splitting completion, Karoubi envelope, tiny objects, essential points), but they all describe the category of bicontinuous functors $Set^A \to Set$. You can find discussion in an article by Borceux and DeJean here; see particularly Proposition 2. 
Question 2: If the tiny or Yoneda objects are dense in $\mathcal{K}$, meaning that the canonical map 
$$\int^{a \in \text{Tiny}(\mathcal{K})} \mathcal{K}(a, k) \cdot a \to k$$ 
is an isomorphism for each object $k$ of $\mathcal{K}$, then $\mathcal{K}$ is equivalent to the presheaf topos $[\text{Tiny}(\mathcal{K})^{op}, Set]$. This result was first given in Marta Bunge's thesis as far as I know. As Mike said, being cocomplete and an atomic category is enough. 
