Are coordination numbers of elements fixed?
No. Coordination numbers of no element are fixed. For example, if you add chloride ions to a pink solution of octahedral $\ce{[Co(H2O)6]^3+}$, a blue, tetrahedral complex is generated according to the following equation:

$$\ce{[Co(H2O)6]^3+ + 4 Cl- &lt;=&gt;&gt; [CoCl4]- + 6 H2O}$$

In fact, if you study the mechanisms of different metal-catalysed organic reactions, you will notice that different coordination numbers of the central metal and specific switching between them is the key to understand these mechanisms.

Finally, for some complexes even establishing their coordination is hard. Zinc(II) forms aquacomplexes like any other cation. But while those complexes of transition metals with any number of d-orbitals unequal to zero or ten are often well-defined and established, that is not the case for zinc. Zinc(II) aquacomplexes rapidly equilibrate from tetracoordinated and pentacoordinated to hexacoordinated species.