Regarding your second question, on other possible deformations of $sl(2)$: 

There have been various studies on (multi-parametric) deformations of Lie algebras -as has already been mentioned in the comments to the OP- during the last decades:     
 An example of a $2$-parameter deformation $sl_{pq}(2)$ which leads to a quantum group is simply given by: 
$$
[H,E_{\pm}]=\pm E_{\pm}, \ \ \ [E_+,E_-]=[2H]_{pq}
$$
where $E_-$ stands for $F$ and the deformation function, in the rhs, is given by: $$[x]_{pq}=\frac{q^x-p^{-x}}{q-p^{-1}}$$ (setting $p=q$ and rescaling the generators produces the $q$-deformation described in the OP as a special case). You can find more details at [arXiv:math/0506539][1], where this deformation is studied and it is proved that it admits a class of infinite dimensional representations with no analogue in the undeformed or in the $q$-deformed case.    
Another -similar- example can be found in the article: [A two-parameter deformation of the universal enveloping algebra of $sl(3,C)$][2], by J.F. Cornwell. A detailed discussion on the hopf structure of the deformed algebra and its implications on the usual hopf structure(s) of the undeformed algebra is also included.  
In [Introduction to quantum algebras][3], by M.R. Kibler, two parameter deformations such as $u_{pq}(2)$ and $u_{pq}(1,1)$ are studied: their hopf algebraic structures are investigated and their realizations (that is: homomorphisms or isomorphisms) with two parameter deformations of the Weyl algebras and the angular momentum algebras are used as a tool of investigating their representations. 


  [1]: https://arxiv.org/abs/math/0506539v1
  [2]: https://link.springer.com/article/10.1007/BF01555521
  [3]: https://arxiv.org/pdf/hep-th/9409012.pdf