Is the questioner considered a pimp for his silence and trust in his sister after he found a letter indicating she was going on a "romantic date" and she denied it, and is his action correct or should he have taken a harsher stance or investigated the matter?

Searching the phone of your wife or anyone else without their permission is impermissible because it is prohibited espionage. What you mentioned about your sister's conversation with your wife is not clear evidence of an illicit relationship, and the default is to assume the best of Muslims. What you mentioned does not place you in a state of dayyatha (cuckoldry). Assume the best of your sister, and beware of spying on her unless clear suspicion arises from her, for searching for faults and tracking down private matters is corrupting, as the Prophet, peace and blessings be upon him, said: "Indeed, if you follow the private matters of people, you will corrupt them, or almost corrupt them."