The answer to the first part is indeed true. In fact, something more general is true. Let $mathcal{A}$ be a small category and let $mathcal{C}$ be a cocomplete category (which is locally small, i.e., there is just a set of morphism between any two objects). Then any cocontinuous functor $L colon mathrm{Set}^{mathcal{A}^mathrm{op}} rightarrow mathcal{C}$ has a right adjoint, given by $C mapsto mathcal{C}(K-,C)$, where $K colon mathcal{A} rightarrow mathcal{C}$ is the composite of the Yoneda embedding and $L$.
This is for example proved in Kelly's "Basic concepts of enriched category theory", Theorem 4.51. He proves the enriched version of this result, where $mathrm{Set}$ is replaced by any complete and cocomplete category $mathcal{V}$. I must say I don't know of a reference that just treats the $mathrm{Set}$-case.
If the target is the category of presheaves on some large category, then this might fail. Take for example $mathcal{D}$ the large discrete category whose objects are sets, and let $F colon mathcal{D} rightarrow mathrm{Set}$ be the canonical inclusion functor. Then the functor $mathrm{Set}rightarrow mathrm{Set}^{mathcal{D}}$ which sends a set $X$ to the functor $Ftimes X$ (i.e., the functor which sends a set $A$ to $Atimes X$) is cocontinuous, because $Atimes -$ preserves colimits. However, there is a proper class of natural transformations $F rightarrow F$ (a natural transformation just amounts to choosing an endomorphism of each set with no compatibility conditions), so if this functor had a right adjoint $R$, then we would have a bijection $mathrm{Set}(ast,RF) cong mathrm{Set}^{mathcal{D}}(F,F)$, i.e., $RF$ would have to be a proper class. The reason for this failure is of course that $mathrm{Set}^mathcal{D}$ is not locally small. Note that this problem doesn't go away when we use universes: the above example would give you an isomorphism between a small set and a large set, i.e., a set outside of the universe.
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