What is it that Hashem remembered about Rachel?
Rashi: Hashem remembered two things - (a) How she handed over the Simanim to her sister 1 and (b) How distressed she was over the fear lest Yakov divorce her due to barrenness, and Esav would marry her, 2 as a result of which He now answered her prayers. 3
On what date did Rachel give birth to Yosef?
Rosh Hashanah, 11a: On Rosh Hashanah. 1
What are the implications of term "Vayiftach es Rachmah"?
Ta'anis, 2a: It implies that the key (Maafte'ach) of childbirth lies exclusively in the hand of Hashem, and has not ben handed over to a Shali'ach.
Why does it say "va'Yizkor" and "va'Yishma"?
Ohr ha'Chayim: Hashem remembers His pious ones even before they pray.
Ha'Emek Davar: She had great deeds; Tefilah does much (the deeds did not suffice by themselves).