Can a Muslim woman who got angry and said, "God does not exist," repent and recite the Shahada to become a Muslim again, and is what she said considered shirk (associating partners with God) even though she said it in a state of extreme anger?

What the questioner said, "cursing the religion," is clear disbelief (kufr sareeh). She must recite the Two Testimonies (Shahadatayn) and intend to enter Islam if she was aware of what she said, for anger is not an excuse unless it completely clouded her mind. The Prophet, peace be upon him, clarified that a Muslim might utter a word that leads him to Hell due to Allah's displeasure. As for supplication (du'a), Allah answers the supplicant in all circumstances: either by granting what is asked, or by averting evil from him, or by storing the reward for the Day of Judgment, as Allah says: ﴿And your Lord said, "Call upon Me; I will respond to you."﴾ [Ghafir: 60]. And it may be that the non-fulfillment of what is requested is better for the supplicant, due to the evil or tribulation it might contain for him, and Allah knows best what is good for His servants.