What kind of role do quantum groups play in modern physics ? Do quantum groups naturally arise in quantum mechanics or quantum field theories? What should quantum symmetry refer to ? Can we say that the "symmetry" of a noncommutative space (quantum phase space) should be a quantum group? Do quantum groups describe "extended symmetry" ?
Yes, quantum groups naturally arise in many physics problems. E.g. solutions of the quantum Yang-Baxter equation appear as scattering matrices of integrable 2-dimensional quantum field theories (see "Quantum fields and Strings: a course for Mathematicians", p.1179). Also, quantum groups appear in the description of monodromy of the vertex operators in the WZW model of conformal field theory ("the Drinfeld-Kohno theorem"). Thirdly, there are spectacular applications of (infinite dimensional) quantum groups to statistical mechanics, which are described in the book by Jimbo and Miwa "Algebraic analysis of solvable lattice models". Also, quantum groups are useful in construction and studying of certain classes of integrable systems (q-Toda systems, Macdonald-Ruijsenaars systems, etc.)
One of the main mechanisms through which quantum groups appear in physics is the same as for usual Lie groups: if a Hamiltonian of a quantum system has a Lie group symmetry then this helps find its eigenvalues and eigenvectors (which is the main problem in studying a quantum system), because its eigenspaces are representations of this group.
It is not quite true that the usual quantum groups are symmetries of WZNW model. The standard root of unity q. groups are hidden symmetries at the level of pre-Hilbert space where some spurious ghost-like norm zero states appear. The true symmetry (in the sense of axiomatic quantum field theory) includes rather certain quotient which is just a weak quasi-Hopf algebra, and whose representation automatically exclude the nonphysical representations with quantum dimension zero. See