Question: What is the original source for this halacha, as it seems from Ezra 2:65, that women and men sung together publicly at the beginning of the Second Temple period.
The Gemorra in Berachos 24A deduces this halacha from the pasuk in Shir HaShirim that lists a melodious voice as an attractive feature. Shulchan Aruch brings this halacha in Even HaEzer 21:1. In the pasuk you quote from Ezra that there were male and female singers to accompany those who travelled from Bavel to build the second Beis HaMikdash it does not say that they sang together, presumably the women singers sang separately to entertain the women travelers.
Question: Is the issur of koil isha also by a sister? R' Moshe and R' Shloime Zalman were matir it, but R' Yaakov Kamenetsky paskined she's the same as any other girl and R' Elyashiv paskined it's only when she's 11 that it's assur. How do we paskin?
Answer: As you mentioned, there are many opinions on the issue. However, it seems to me that the prevalent custom is to be lenient until she marries. Until she moves out of the house it is likely that the brothers are familiar with her voice, which would decrease after she moves out.
Question: I work as an occupational therapist with special needs children and often use songs as a way of communicating with them. A male therapist was just hired, and I was wondering if I am permitted to sing to the children while he is in the room? Thank you.
As long as you make the male therapist aware that your singing is for therapeutic purposes and not for personal enjoyment it is permitted. The rationale is that many poskim hold that the heter of "tarid b'melachto" applies under these circumstances you are not committing "lifnei iver" by causing him to enjoy your voice' since the male OT is involved in his work related responsibilities and is not m'chavin for hanaah. This same heter is used in many schools with male principals who often observe classrooms. If the male therapist is not Jewish then lifnei iver would not apply in this case and it would not even be necessary to inform him of your motives.