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" ?



What is the  relation between quantum symmetry  and  quantum groups?
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. 


