Q1 asks two very different questions: the answer to the first one is definitely not, and the answer to the second one is almost.
Let me start with the second one. First of all, to handle infinite dimensional examples like $U_q(\mathfrak g)$ (which, strictly speaking, is not a ribbon Hopf algebra), as usual with Tannaka duality it's better to consider instead (finite-dimensional) comodules over a co-ribbon Hopf algebra. Then the general slogan is that the "hard" part is to reconstruct $A$ as a bialgebra, and then the extra structures or properties (Hopf, braided, ribbon,..) come pretty much for free. Still, knowing the underlying monoidal category is not enough, you also need to have the monoidal forgetful functor to vect.
So at the end of the day I'd argue your question has little to do with the ribbon structure, and basically boils down to whether all monoidal categories come from bialgebras, which is well-known to be false. The most obvious example in the ribbon case is of course $FrTang$ itself. There are also lots of examples which are full sub-categories of ribbon categories of the form you want, without being of this form themselves. Even if you do restrict to categories which you know are categories of (co)modules, there are lots of weakenings of the axioms of Hopf algebras (e.g. quasi Hopf, weak Hopf, Hopf algebroids, etc..), and in all of theses cases it makes sense to have a ribbon structure.
For the second question, if $q=e^h$ for a formal variable $h$ (which is the only thing that makes sense for arbitrary Lie algebra) then those are essentially classified by formal solutions $r \in \mathfrak g^{\otimes 2}[[h]]$ of the classical Yang-Baxter equation, for which $r+r^{2,1}$ is $\mathfrak g$-invariant. If you want $q$ to be a variable, there are some known results for reductive Lie algebras, see e.g. https://mathoverflow.net/a/314487/13552.