Does the white dwarf star rotate?
Yes - All stars rotate, including white dwarfs. In fact compact objects ( white dwarfs, neutron stars, and blackholes ) rotate faster than main sequence stars.
It is very unusual to come across a non-rotating star. Star form through the gravitational collapse of cold molecular gas and dust (star forming nebula). Such star forming nebulae almost always have some angular momentum. As the nebula collapses, its angular momentum remain conserved in most cases (because there is negligible external torque in most cases).
Angular Momentum Conservation implies that a collapsing mass should start spinning faster and faster. The core of the collapsing nebula which forms the star is already spinning when it forms.
Sun for example is spinning at a rate of approximately 25 days.
A star like our Sun which is in its main sequence phase is supported against gravitational collapse by radiation pressure. But when such stars reach the end of their main sequence stage after loosing all of their nuclear fuel the radiation support is withdrawn and gravitation collapse the star further. The star implodes. An imploding star would start spinning crazily to conserve angular momentum.
So almost all compact objects will be found rotating at very high speeds. For example some Pulsars which are neutron stars have rotation time periods in milli-seconds. When they are normal stars they would have had rotation time periods similar to our sun.