I say, and with God's success: It is permissible to perform ablution and bathing with it, and the water remains pure and purifying for the event; because the frog is an aquatic-born animal, unlike if an animal that was born outside of water dies in the water, it is not permissible to perform ablution with it, and the water becomes impure by its death in it, even if it lived in the water; because the criterion is birth in water, not living in it. Refer to: Al-Wiqayah and its explanation by Sadr al-Shari'ah, pages 96-97, and God knows best.