Closed monoidal structure on the derived category of sheaves

The most direct answer to question (2) is no, even in the case of sheaves on a point. In this case we simply have a ring R and we are asking whether the derived category of R-modules is equivalent to the category of R-module objects in the derived category of abelian groups. For example, let $R = Z/p^2$. Then in the derived category of R-modules, we have nonzero Ext groups
$$Ext^n_{Z/p^2}(Z/p,Z/p) = Hom_{D(Z/p^2)}(Z/p,Z/p[n])$$
for all positive degrees $n$. However, because no object in the derived category of Z-modules has any nontrivial Ext-groups in high degrees, we have for $n$ large
$$Hom_{dMod(Z/p^2)}(Z/p,Z/p[n]) = 0.$$
The problem is that dMod(R) doesn't really remember that R is an actual ring, but instead only remembers that it can be lifted to a chain complex with a chain-homotopy associative multiplication map. These kinds of ring objects are too weak to do really serious homological algebra and define a proper derived category.

The answer to question 3 is correspondingly no, because we can't really construct proper derived functors of the tensor product given only a monoid in Der(X).