Do non-redox reactions exist?
There are many, here are some examples


Ion interchange reactions of various types, such as precipitation

$\ce{NaCl + AgNO3 -&gt; AgCl \downarrow + NaNO3 }$
Polar molecule insert/ejection. The most common case is hydrolysis and reverse reactions.

$\ce{PCl5 + 4 H2O -&gt;  H3PO4 + 5 HCl}$
Reorganization of bonds between atoms of same type, such as catalytic benzene synthesis from acetylene 

$\ce{3 C2H2 -&gt; C6H6}$
Ligand exchange

$\ce{ [Fe(H2O)6]^{3+} + 6CN- -&gt; [Fe(CN)6]^{3-} + 6 H2O}$
polar oligomerization

$\ce{ H2O + n CH2O -&gt; HO-(-CH2-O-)_{n}-H}$