Is there an example of a ring which has weak global dimension 2 and is not coherent? 
Is there an example of a ring  which has (weak) global dimension 2, and is not coherent? 

We know that there exist coherent rings with weak global dimension 2  there are also Noetherian rings of global dimension 2.
 A: Perhaps you want a commutative example. If not, there are fairly simple non-commutative ones.
Let $Q$ be the quiver with three vertices, one arrow $\alpha$ from vertex $1$ to vertex $2$, and infinitely many arrows $\{\beta_s\vert s\in S\}$ from vertex $2$ to vertex $3$, and let $R$ be the quotient of the path algebra of $Q$ over a field $k$ by the ideal generated by $\{\alpha\beta_s\vert s\in S\}$.
More explicitly, $R$ has a basis $\{e_1,e_2,e_3,\alpha,\beta_s\vert s\in S\}$, with multiplication of basis elements given by


*

*the elements $e_i$ are orthogonal idempotents (i.e., $e_i^2=e_i$ and $e_ie_j=0$ for $i\neq j$)

*$e_1\alpha=\alpha=\alpha e_2$

*$e_2\beta_s=\beta_s=\beta_s e_3$ for all $s\in S$

*all other products of basis elements are zero.
Then $R$ has right global dimension and weak global dimension equal to $2$, and is not right coherent:
It's not right coherent since the right ideal generated by $\alpha$ is finitely generated but not finitely presented.
Let $J$ be the two-sided ideal spanned by the arrows. Then $J^2=0$ and $R/J$ is a semisimple algebra isomorphic to $k\times k\times k$. So every right $R$-module $M$ is an extension of a direct sum $M/MJ$ of simple $R/J$-modules by another direct sum $MJ$ of simple $R/J$-modules.
So the global dimension of $R$ is the supremum of the projective dimensions (as $R$-modules) of the three simple $R/J$-modules, which is easily calculated to be $2$.
Similarly, the weak global dimension of $R$ is the largest $d$ such that $\operatorname{Tor}^R_d(S,T)$ is non-zero for some simple (right and left) simple modules $S$ and $T$. But $$\operatorname{Hom}_k\left(\operatorname{Tor}^R_d(S,T),k\right)\cong
\operatorname{Ext}_R^d\left(S,\operatorname{Hom}_k(T,k)\right),$$
so this is the same as the global dimension.
