What is the correct and practical way to remove the impurity of madhiy (pre-seminal fluid) and wadiy (thick white discharge) from clothes? Is it sufficient to just pour water over it, or is scrubbing required? And is it necessary to recite the shahada (testimony of faith) inwardly while purifying it?

It is neither obligatory nor recommended to pronounce the Shahada, whether by heart or tongue, when pouring water on a garment. Believing that this is obligatory or recommended is an innovation (bid'ah). One can find out how to remove madhiy and wadiy from clothes, and whether it is by sprinkling or washing, by referring to other fatwas.