In the situation you have described, if the accident happened through your fault and the woman's car was damaged, then compensating that damage is your responsibility under Shariah. The answers to your questions are as follows:
1) The insurance company has already covered the woman's loss. However, she had taken out an insurance policy on her car and would have been paying regular premiums for it. Paying for the actual damage therefore remains your responsibility.
As a general rule, the compensation should be paid to the woman herself. However, if she has authorised the insurance company to act as her agent in claiming it, then the company may also demand payment, and you may pay the company instead.
2) You are only responsible for the actual damage. If the company or the woman demands more than that, you have the right to negotiate a fair amount.
3) If paying the full amount at once would cause you financial difficulty, it is permissible to ask the woman or the insurance company for a reduction, more time, or payment in instalments. A settlement for a lesser amount is also valid if both parties agree to it.
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دلائل:
رد المحتار على الدر المختار: (181/6، ط: دار الفكر)
مَنْ أَتْلَفَ مَالَ غَيْرِهِ بِغَيْرِ إِذْنِهِ ضَمِنَهُ
بدائع الصنائع: (164/7، ط: دار الكتب العلمية)
وَأَمَّا الضَّمَانُ فَيَجِبُ بِإِتْلَافِ مَالِ الْغَيْرِ، لِأَنَّ الْإِتْلَافَ سَبَبُ وُجُوبِ الضَّمَانِ
مجلة الأحكام العدلية: (ص: 303، المادة: 1552)
إذَا صَالَحَ أَحَدٌ عَنْ دَيْنِهِ الَّذِي هُوَ فِي ذِمَّةِ الْآخَرِ عَلَى مِقْدَارٍ مِنْهُ يَكُونُ قَدْ اسْتَوْفَى بَعْضَ دَيْنِهِ , وَأَسْقَطَ الْبَاقِيَ أَيْ أَبْرَأَ ذِمَّةَ الْمَدِينِ مِنْ الْبَاقِي
و فیه ایضا: (ص: 303، المادة: 1553)
إذَا صَالَحَ أَحَدٌ عَلَى تَأْجِيلِ وَإِمْهَالِ كُلِّ نَوْعٍ مِنْ مَطْلُوبِهِ الَّذِي هُوَ مُعَجَّلٌ يَكُونُ قَدْ أَسْقَطَ حَقَّ تَعْجِيلِهِ
Allah the Almighty knows best.
Darul ifta Al-Al-Ikhlas, Karachi.