I understand abortion in general is a sin as per the answers to this question here
But what if mother's life is in danger due to a complicated pregnancy. In that case is it permissible ? If not mistaken saving one's own life allows for transgressions as per Apaddharma. So same applicable in this case ?
What about the surgeon or doctor who does such an act. While mother may have a valid reason can the same be extended to the one performing this act ?
It's important to note that there's clear distinction in Saṁskṛta b/w the terms for abortion and miscarriage, the former being deliberate termination of pregnancy while the latter being involuntary termination of pregnancy . Abortion is usually called garbha- (hatyā/vadha) or bhrūṇa (hatyā/vadha), and miscarriage is known by names such as - sraṁsana (as in Gautamadharmasūtra 2.5.15), or garbha- (srāva, pāta,..). In Mitākṣarā on Yājñavalkya smṛti (3.20), Vijñāneśvara states that - miscarriage is called srāva till the 4th (month of pregnancy), pāta for the 5th & 6th (months); prasūti from the 7th (month) onwards, while in the 10th (month) it's sūtaka. (ā caturthād bhavet srāvaḥ pātaḥ pañcamaṣaṣṭayoḥ / ata ūrdhvaṁ prasūtiḥ syād daśāhaṁ sūtakaṁ bhavet). Although abortion is a pāpa yet miscarriage is not so, as per śāstras. However, a mother does enter aśauca as a result of miscarriage and has to undergo śuddhi (garbhamāsasamā rātrīḥ sraṁsane garbhasya - Gautamadharmasūtra 2.5.15).
Suśruta saṁhitā cikitsāsthāna (15.3-11), supports abortion when the life of the mother is at stake and endeavors to protect the life of the mother at all costs, in such critical case. However, Suśruta doesn't go into a dharmaśāstrika discussion of whether such an act would be pāpa (or not) on the part of the vaidya & mother, under the said circumstances. Instead, he takes up this issue from an Āyurvedic perspective. So, even if dharmācārya(s) disagree with his opinion on this matter, what Suśruta says (here) would be valid for a medical practitioner, because he's taking the recourse to abortion only under extreme circumstances, and not otherwise.
Also, Suśruta saṁhitā is one of the foremost prāmāṇya Āyurveda (cikitsā)śāstras. As per sūtrasthāna (1.1-2), Dhanvantarī himself took avatāra as son of Dhanvā and his descendant Kāśirāja Divodāsa gave upadeśa on Āyurveda to R̥ṣis such as Aupadhenava, Vaitaraṇa, Aurabhra, Pauṣkalāvata, Karavīrya, Gopurarakṣita, Suśruta, etc.
Āyurveda is also an upāṅga of Atharva Veda (sūtrasthāna 1.6).
Suśruta says, "There's nothing as difficult as the delivery of a fetus astray in the womb, for her... the job must be done 'by feel' ... by one hand, without injury to mother or fetus (if possible). (nāto'nyāt kaṣṭamam .....cāhiṁsatā - 15.3). He continues : "If the fetus is alive, one should attempt to remove it from the womb of the mother (alive)" (jivati tu garbhe sūtikā garbhanirharaṇe prayateta - 15.5). No doubt is left as to the ideal to be striven for : the safety of both mother and child. However, if the fetus is dead (mṛte garbhe), it may be removed by cutting (and disembering, if necessary), as per 15.9. The śāstra then considers the situation in which the live fetus cannot be safely delivered. In this event, it forbids removal by surgery, "For if (the fetus) be cut one should harm both the mother and the offspring" (dāryamāṇo hi jananīm ātmānaṁ civa ghātayet / aviṣahye vikāre tu śreyo garbhasya pātanaṁ / na garbhiṇyā viparyāsas tasmāt prāptaṁ na hāpayet- 15.10-11). In an irreedeemable situation, it's best to cause the miscarriage of the fetus, for no means must be neglected which can prevent the loss of the mother (and protect her from harm). Abortion then is the last recourse when it's clearly a question of weighing life against life - the life of the mother against that of the fetus.
(Lipner, 1989, pg. 49-50).
In order to answer this question let us take a look at why dharma is needed in Samsara. Why can't we just have artha and kama as goals in samsara?
Bhishma on Dharma
Bhishma said, ‘..it is difficult to say what righteousness is. It is not easy to indicate it. No one in discoursing upon righteousness, can indicate it accurately. Righteousness was declared (by Brahman) for the advancement and growth of all creatures. Therefore, that which leads to advancement and growth in righteousness. Righteousness was declared for restraining creatures from injuring one another. Therefore, that is Righteousness which prevents injury to creatures. Righteousness (Dharma) is so called because it upholds all creatures. In fact all creatures are upheld by righteousness. Therefore, that is righteousness which is capable of upholding all creatures. Some say that righteousness consists in what has been upheld in the Srutis. Others do not agree to this. I would not censure them that say so. Everything, again, has not been laid down in the Srutis.’
[This refers to the well-known definition of Dharma ascribed to Vasishtha, viz, “That which is laid down in the Srutis and Smritis is Dharma.” The defect of this definition is that the Srutis and Smritis do not include every duty. Hence Vaisishtha was obliged to add that where these are silent, the examples and practices of the good ought to be the guides of men, etc.]
Mahabharata, Santi Parva, Section CIX
Dharma is necessary to prevent injury to creatures and to uphold all creatures.
How do we apply dharma in the case of a mother giving birth to the child? The first priority is of course to save the lives of both mother and baby.
What should we do when the doctor thinks that only one life can be saved either that of the mother or that of the baby?
Obviously from the point of dharma we have to make sure that our decision is not cruel to either the mother or the baby. At the very least cruelty should be minimized. This is also the spirit of the three quotes given below.
However, discard the desire (kama) and material wealth (artha) if contrary to Dharma; as also, any usage or custom or rules regarded as source of Dharma if at any time they were to lead to unhappiness or arouse people's indignation.
Manu Smriti 4.176
When must holy acts not be done?
If a holy act is against the interest of other members of the society, it should not be practiced. It is Dharma which is the source of Artha and even of Kama.
Kurma Purana I.2.54
Attitude towards ancient custom
One should practice what one considers to be one’s duty, guided by reasons, instead of blindly following the practices of the world.
Mahabharata Santi Parva Section CCLXII
So the question boils down to whose death will be more cruel, that of the mother or that of the baby. Who will suffer more? How do we minimize cruelty? The answer is that the mother's life will get priority over that of the baby. The reason is that the mother will suffer more when she realizes that she will die than the baby.
As you have linked in the question we can see that in Hinduism abortion is a crime, but those, as you have pointed out do not say anything about what to do when the woman's life is in danger.
As I have not been able to get the prescription of action when there is a danger of either baby or the woman from the shastras, I'll try to use the overall meaning and spirit of the shastras to understand the issue.
Mahabharata says 2.3.10
त्यजेत् कुलार्थे पुरुषं ग्रामस्यार्थे कुलं त्यजेत् |
ग्रामं जनपदस्यार्थे आत्मार्थे पृथ्वीं त्यजेत् ||To save a family, abandon a man; to save the village, abandon a family; to save the country, abandon a village; to save the soul, abandon the earth.
Shiva Purana, Uma Samhita, Chapter 20
मानुष्यं च समासाद्य स्वर्गमोक्षप्रसाधनम्। नाचरत्यात्मनः श्रेयः स मृतः शोचते चिरम्॥३०॥
The human body is the means for achieving of the heaven and moksa and even after achieving the same, he does not do good to himself. After death he keeps on thinking for a long time.
देवासुराणां सर्वेषां मानुष्यं चातिदुर्लभम्। तत्सम्प्राप्य तथा कुर्यान्न गच्छेन्नरकं यथा॥३१॥
It is difficult for even the gods and the demons to achieve the human body. Therefore after achieving the human body, one should act in a way that he has not to face the agonies of the hell.
From this we can understand that human birth is very rare and it is precious. And it is the reason for Moksha. So for the soul even the Earth can be sacrificed.
And we know human birth is very precious and rare and you cannot lose it. So one should save one's life which is precious. So the mother should always save her own life.
This is not same as normal late term abortion, that his himsa. But when you abort the baby when the mother life is in danger is dharma himsa, self defense.
So to save one's life, especially when human birth is so rare and in self defence even wars can be fought, so isn't it clear to save the life of a woman? Ofcourse this doesn't apply to casually having abortions for career etc. It is only when the life of mother is in danger.
The answer can be both Yes and No.
In the book - Dancing with Siva: Hinduism's Contemporary Catechism by Satguru Sivaya Subramuniyaswami (2003 Edition), the acharya says -
Mandala 15: Husband and Wife
Hinduism does not exclude or draw harsh conclusions against any part of human nature, though scripture prohibits adultery and forbids abortion except to save a mother's life. Advice in such matters should be sought from parents, elders and spiritual leaders. The only rigid rule is wisdom, guided by tradition and virtue. The Vedas beseech, "May all the divine powers together with the waters join our two hearts in one! May the Messenger, the Creator and holy Obedience unite us." Aum Namah Sivaya.
(page 89)
In this excerpt from the Padma Maha-Purāṇa, we get following words from Rishi Vishvamitra, who's ready to sacrifice himself for a Child's life.
इति तेषां वचः श्रुत्वा स विप्रः सदयोऽभवत् ॥ ३८ ॥ प्राणा ममापि गच्छंतु सुखी भवतु बालकः । बालकार्थे द्विजार्थे च स्वाम्यर्थे ये जना इह ॥ ३९ ॥ त्यजन्ति तृणवत्प्राणांस्तेषां लोकाः सनातनाः । विमृश्येति मुनिः स्वांते स प्रोवाच द्विजर्षभः ॥ ४० ॥
Hearing these words of them, the brāhmaṇa (i.e. Viśvāmitra) became compassionate. ‘Let even my life depart; (but) the boy be happy. Those people who in this world cast their life like (a blade of) grass for a child, or a brāhmaṇa, or their master, obtain eternal worlds.’
Now here, it's a grown-up Child, for whom Vishvamitra is ready to sacrifice himself, but the killing of foetus is said to be an unforgivable sin, so basis the above excerpt, it may also be concluded that whatever be the circumstances, abortion is not permissible.
ŚB 1.8.10, Śrīmad-Bhāgavatam Mahā-Purāṇa
अभिद्रवति मामीश शरस्तप्तायसो विभो । कामं दहतु मां नाथ मा मे गर्भो निपात्यताम् ॥
O my Lord, You are all-powerful. A fiery iron arrow is coming towards me fast. My Lord, let it burn me personally, if You so desire, but please do not let it burn and abort my embryo. Please do me this favor, my Lord.
So, as per this view, everyone involved in the process of abortion, including the doctors incurs sinful Karmic fruit (cf. Manu Smriti Verse 5.51), since we have established abortion to be a sin, be it any circumstances.