Shabbat 36: Naming Objects; The Stove on Shabbat & 'Women's Work'

Are we permitted to move a shofar inside the home on Shabbat?  The rabbis wonder whether shofarim are the same things as trumpets.  They debate about whether these are two different words for the same object, used at different points in time.   It is noted we are faced with similar conundrums with other words.  

What difference does it make whether a word is changed if the meaning is the same?  We are told that get, a woman's divorce paper, contains the name of the place where it was written.  The name of that place cannot change for it would invalidate the get.  

We are introduced to Perek III of Massechet Shabbat in amud (b) of today's daf.  The Mishna teaches that if a stove was already lit with straw or rakings on Shabbat, we are permitted to place a pot of cooked food on top of the stove.  Anything that could alight on Shabbat must be removed from the stove.  Beit Shammai say that only hot water can be placed on the stove because the food could cook further and because one might be tempted to stoke the fire.  Beit Hillel say that both hot water and cooked food are permitted.  Beit Shammai add that one may remove a pot from the stove but not replace it.  Beit Hillel say one may return the pot to the stove.

Work done inside the home on Shabbat is particularly of interest to women, who were said to be in charge of the halachot inside of the home.  Women have always been expected to do the less valued work of cooking, cleaning, childcare. To trust women with the observance of  Shabbat-based halachot is significant.  Why would it be that women can be trusted with these halachot but we cannot be trusted with positive/time-based mitzvot?  Again, it seems that halachot are often based on what our Sages determined to be most functional; what would maintain their current organizational structure that maintained the power of men and (in particular) of Torah scholars.