1) Scholars say that Imam Hussayn told his sister that because he did not want her to lament at the moment of his martyrdom. He wanted Sayeda Zainab to show strength so the enemies would not deem the Ahlulbayt as weak and humiliate them.
However after Ashura, there is no problem with mourning. We have many traditions that encourage people mourn Imam Hussayn.
2) All of the Imams did. We have several hadith that the Prophet lamented in sunni books.
Ahmad Ibn Hanbal narrates the hadith of Um Salamah.
As for the Imams, we have multiple hadith of Imam Sadiq, Imam Rida with Dibil al-Khuzai.
This book is a good source