The following is not really an answer but a rather too-long comment, with respect to your second question:

Does there exist deformations of the monoidal category of $U(\frak{g})$-modules which are not equivalent to the cateogry of modules of $U_q(\frak{g})$, for some $q$?

2-parameter deformations of the universal enveloping algebras $U_{pq}(\frak{g})$ have been studied such as for example: $U_{pq}[sl(2)]$, $U_{pq}[sl(3,C)]$, $U_{pq}[u(2)]$, $U_{pq}[u(1,1)]$, etc. Most of these are based on the two-parameter deformation function
$$
[x]_{pq}=\frac{q^{x}-p^{-x}}{q-p^{-1}}
$$
I am not really sure as to whether the representation categories of such deformed algebras are braided or even monoidal or if they may be described as deformations of the symmetric monoidal categories of the representations of $U(\frak g)$.

However, in the first of the above references, the authors claim that:

the two-parametric quantum group denoted as $U_{pq}[sl(2)]$ admits a class of infinite-dimensional representations which have no classical (non-deformed) and one-parametric deformation analogues, even at generic deformation parameters

which may be considered -in my understanding- as an indication that the representation categories of $U_q[sl(2)]$ and $U_{pq}[sl(2)]$ cannot be equivalent.

Thus, if i am correct in this and given the excerpt from the OP, the categories of $U_{pq}[sl(2)]$-modules cannot be braided.

Also, given Noah Snyder's answer, such categories of representations of two-parameter deformed algebras cannot even be monoidal.

On the other hand, in the last of the above cited articles, a $pq$-deformed coproduct $\Delta_{pq}$ is given for $U_{pq}[u(2)]$ and an element $R_{pq}$ such that:
$$
\Delta_{qp}=R_{qp}\Delta_{pq}R_{qp}^{-1}
$$
For $p=q^{-1}$, $R_{qp}$ reduces for the universal $R$-matrix of Drinfeld. However, i do not generally know if categorical implications (such as an associated braiding or deformed braiding) have been explicitly studied for $R_{qp}$ and more generally for deformations of this kind.