Is it permissible to get rid of a paralyzed cat, unable to move, that I found in the street, knowing that my financial situation does not allow for its treatment, and I cannot keep it in the house for fear of impurity and transmission of diseases, along with the inability to leave it in the street due to its inability to defend itself or search for food?

Your act of saving the cat is commendable, and you will be rewarded for it, based on the hadith of the Messenger of Allah, peace and blessings be upon him: "A prostitute woman was forgiven. She passed by a dog at the top of a well, panting. He said, 'Thirst was about to kill it.' So she took off her slipper, tied it with her head covering, and drew water for it. She was forgiven for that." There is no religious impediment to removing a cat from the house if its presence causes harm or annoyance. It can find food outside or someone to care for it. It is not permissible to kill it without a reason or with the intention of showing mercy to it.