Who is the woman who talks to young Tooru Amuro/Rei Furuya?
You are talking about episode 771 of the anime (chapter 892 in the manga) The woman was Elena Miyano, the mother of  Shiho Miyano (Ai Haibara) who created the first prototipes of the APTX-4869 for the BO. 

Acording to DC world wiki:


  While the true relationship between them has not been revealed yet, We know that they knew each other when Rei was younger. Rei got into many fights and it seems that Elena would bandage him up after the fights. However, one day, Elena had to move and could no longer bandage 


Amuro's real name is Rei Furuya and "Rei” (零) means “zero” and that is possibly the reason why he is called zero.