Necessary and sufficient conditions for all sheaves on a site to be continuous functors? All representable functors are continuous. This makes it possible to associate additional natural operations with them, which are absent for arbitrary presheaves.


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*What are the sufficient and what are the necessary conditions on the subcanonical site $I$ for all sheaves to be continuous?

*Is there a canonical way to introduce the structure of such a site on an arbitrary small category? (by canonical I mean functorial with respect to category equivalences)


 A: Given a category $C$, and a familly of co-cone in $C$ (you can take all colimit cocone in $C$ if you want - the family doesn't even have to be small) there is a smallest topology on $C$ so that sheaves for this topology sends these cocone to limit cones.
The detail of the construction below should also give some answer to your question 1.
For each of the special cocone $F:I^\triangleright \to C$, we consider the map $colim_I F \to F(*)$ in the category $P(C)$ of presheaves over $C$. A presheaf sends this cocone to a colimit if and only if it is right orthogonal to this map. A Grothendieck topology is the same as a left exact localization of $P(C)$, so we are looking for the smallest left exact localization that inverse these maps.
A Grothendieck topology invert a map $f:X \to Y$ if and only if it invert the two monomorphism $Im(f) \to Y$ and $X \to X \times_Y X$ and it inverts a monomorphism $A \hookrightarrow X$ if and only if for each representable $c$ and each map $c \to X$ (that each element of $X(c)$) the Sieve on $c$ obtain as the pullback of $A \to X$ is a covering sieve.
So, putting everything together, saying that some class of maps $colim_I F \to F(*)$ is inverted by the topology (that is that sheaves sends the corresponds cocone to a limit cone) can be written as a fact that a bunch of sieve are covering sieve. and hence there is a smallest topology which satisfies these conditions. (the Grothendieck topology generated by those sieve).
Of course this topology has no reason to be subcanonical in general. For example, assuming that the category $C$ has pullback, this topology being subcanonical implies in particular that colimits in $C$ are universal (and this is probably not sufficient).
