1) The source for this is the collection of reports we have. I am unable to reference a report right now, however, but I thought this was well known. If you think I am mistaken, feel free to correct me.
2) No. But the difference you mentioned, don’t you think it makes no sense in this context whatsoever? I think either the words have been used interchangeably or دار has been used as a neighbourhood.
3) Good point, but no, I don’t think so. The reason being that men attended the Prophet and could attend the Prophet more often than women. Naturally, it was almost impossible for women to be more knowledgeable than men. The women who could be more knowledgeable than men were naturally around the Prophet and that meant that the Prophet was himself there to lead the prayers. Umm e Waraqa may have hailed from an area far from the Prophet and the men there may not have been learned or the learned may not have been present, instead travelling for preaching others. What I mean to say is the chances of women being more learned than men were almost zero. That is one of the reasons women did not lead prayers.
On the other hand, if we take this report to mean that the Prophet appointed her to lead her household, why didn’t he order other women to lead their households? Of course, there must be many women around to be led in their own households, don’t you think? It isn’t like we are talking about mosques in this interpretation that there can only be one in an area. Why just one? And why “order” her?
Also, based on Arab culture and what concessions Islam offered women, it seems they weren’t particularly comfortable in the presence of men. Remember that Islam allows them the concession not to raise their voice to alert the imam of a mistake. So, when learned men are available, why would women lead prayers? Also, I think etiquettes dictate that men lead the prayers. However, this may have not been the case with Umm e Waraqa. Hence, the Prophet “ordered” her to lead the prayers (she was the only one available).
Would love to hear your thoughts!