Do women miss the time for the opening takbir if they begin the pre-prayer sunnah (optional) prayers immediately after the adhan and the prayer takes more than fifteen minutes?

Your beginning the Sunnah prayer immediately after the Adhan is correct and is the preferred approach. You should then follow it directly with the obligatory prayer if you are praying alone at home. Initiating the prayer at the beginning of its time is better.

The time of Takbirat al-Ihram for you is the moment you begin your prayer if you are praying alone, and a person praying alone has no relation to anyone else's Takbirat al-Ihram. As for women who pray in the mosque, the time of Takbirat al-Ihram for them is the time their Imam performs his Ihram.

Attaining Takbirat al-Ihram with the Imam is by performing Ihram with him after his Ihram and before he begins reciting Al-Fatiha.