What is the ruling concerning a garment that has been afflicted by an impurity, such as "splashes of urine or droplets of impure water," and was then purified merely by immersion in water, given that the person who did this was suffering from severe obsessive-compulsive disorder? And is it obligatory to wash the places she sat on while wet with that garment?

You are not required to wash the garment if you are not certain that it has been contaminated by impurity, especially given the presence of waswas (obsessive whispering). The rule "certainty is not removed by doubt" means that a ruling of impurity is only made with certainty. Your garment is pure, and its contact with other things does not render them impure. Continue to disregard the whisperings of Satan.