Is my understanding correct that we don't know the gender of उसने here? Since the verb पढ़ी takes the gender of चिट्ठी? In other words, would "He read your letter" also be correct?
Yes...but he wasnt in the option thats why we will have to use she
Yes that was my answer and it was accepted
Can चिट्ठी also mean "note"? (As in "The teacher didn't see as my note to Jenny passed through the classroom.")
Yes. Interestingly, such notes are called 'chits' in Indian English.