Is it permissible to search and read a sister's chat logs without her knowledge, and to inform the mother of her suspicious appointments to deter her, and do these actions fall under the Prophet's (peace be upon him) prohibition of suspicion and spying, and the Quran's prohibition of spying and backbiting?

Lesbianism is forbidden by scholarly consensus, and Ibn Hajar considered it a major sin. Espionage is originally forbidden, but it is permissible to spy on suspicious individuals to prevent evil. If it is highly probable that a forbidden act will be committed, resulting in the loss of an inviolability that cannot be redressed, then espionage and disclosure are permissible.