What is the ruling on discarding clothes that contain drawings and images, and that were touched by a cat suspected of being infested with ticks? And does the questioner bear a sin for doing so?

There is no need to be preoccupied with obsessive thoughts. Washing clothes is sufficient to purify them from insects. Giving clothes to cats is futile and a waste of money, which is forbidden by Islamic law. Instead, they can be given as charity to those in need. Wearing clothes that have images of animate beings is a matter of scholarly disagreement. Even according to the view that wearing them is forbidden, it is not permissible to destroy them. Rather, they can be put to good use by making them into pillows. Some scholars have explicitly stated the prohibition of destroying something that is fit for use, especially if there are people who can make use of it.