Muslim’s don’t observe any kind of sabbath. We’re forbidden from working on Fridays, as per the command in surah al Jummuah [surah 62:9], but that’s not the same as the sabbath that the jews observed. On Fridays, muslims go to their local mosque, sit on the floor and wait until the khatib [preacher] arrives. The call to prayer is anounced and then the khatib preaches a sermon. After that, a second call to prayer is announced and everyone gets up and prays in unison.
Yes, men and women are segregated during the prayer. That was the practice of prophet Muhammad and his community, so we do the same. A child is permitted to be with a parent of the opposite sex, though. A woman can pray with her child, irrespective of the child’s sex; likewise, a man can pray with his child, irrespective of the child’s sex.
There’s no singing involved, but the Qur’an is recited during the prayer; and if you’ve ever listened to a Qur’an recitation before, you’ll know that it’s recited in rhythmic tones, which is somewhat similar to singing. After the prayer is done, you’re free to leave, donate money, chill out and read a book, make optional devotional practices, etc.