Adi Sankara taught that God incarnated. In his Introduction to his commentary on the Gita, Sankara says (Bhagavad Gita with the commentary of Sankaracarya, Swami Gambhirananda translator, pp 3-5):
When, after a long time, dharma became overpowered by adharma (vice), and adharma increased owing to the deterioration of discriminative knowledge, caused by the rise of desire in the minds of the followers (of this dharma), then, as tradition goes, Visnu, called Narayana, the Prime Mover, took birth--as part of Himself--as Krsna, the son of Devaki by Vasudeva, for the protection of Brahminhood which is Brahman manifest on earth, and for ensuring the stability of the world...
And He, the Lord, endowed with Knowledge, Sovereignty, Power, Strength, Valour and Formidability, exercises His command over His own Maya which naturally belongs to (Him as) Visnu, and which goes by the name Primal Nature...and as such, through His own Maya, He appears as if embodied, as if born, and as if favouring people--though by His nature, He is birthless, changeless, the Lord of all creatures, eternal, pure, conscious and free.
and further in the same book Krishna says in the Gita IV. 7-9:
O scion of the Bharata dynasty, whenever there is a decline of virtue and increase of vice, then do I manifest Myself.
For the protection of the pious, the destruction of the evil-doers, and establishing virtue, I manifest Myself in every age.
He who thus knows truly the divine birth and actions of Mine does not get rebirth after casting off the body. He attains Me, O Arjuna.
Adi Sankara makes no comments on these verses in his commentary. He accepted Krishna's words regarding incarnations.
Now regarding your comments on Paramukta and Jivamukta. Either you are misinterpreting Yogananda or he has taken some license in his interpretation. The Brahma Sutras define jivamukta in 4.1.13-14 (Adhikarana 9). It says and Sankara's commentary follows (Brahma Sutras according to Sri Sankara, Swami Vireswarananda translator, pp 418-420, also available here - http://www.wisdomlib.org/hinduism/book/brahma-sutras/d/doc78348.html):
- When that (Brahman) is realized (there result) the non-clinging and destruction of the subsequent and previous sins respectively, because it is (so) declared (by the scriptures).
[Sankara's commentary] The state of Jivanmukti is described here. The opponent holds that Liberation is attained, in spite of Knowledge, only after one has experienced the results of one’s sins committed before illumination. For the Smritis say, “Karma is not destroyed before it has given its results.” The law of Karma is inexorable. This Sutra says that when a person attains Knowledge, all his past sins are destroyed and future sins do not cling to him. For by realizing Brahman he experiences that he never was, nor is, nor will be an agent, and such a person cannot be affected by the result of sins. The scriptures also declare that. “Just as cotton growing on reeds is burnt when thrown into fire, even so are burnt the sins of one who knowing this offers Agnihotra” (Chh. 5. 24. 3); “The fetters of the heart are broken, all doubts are solved, and all works are destroyed when He who is high and low is seen” (Mu. 2. 2. 8); “As water does not wet the lotus leaf, even so no sins cling to him who knows it” (Chh. 4. 14. 3). What the Smritis say about the inexorability of the law of Karma is true only of ordinary people, and does not hold good in the case of the knowers of Brahman. And in this way alone can Liberation result—by snapping the chain of work. Otherwise Liberation can never take place.
- Thus there is non-clinging of the other (i.e. virtue) also; but at death (Liberation i.e. Videhamukti is certain).
[Sankara's commentary] As a knower of Brahman has no idea of agency he is not affected by good deeds also. He goes beyond vice and virtue. “He overcomes both” (Brih. 4. 4. 22). And as he is not touched by vice or virtue after illumination, and as his past sins are destroyed by Knowledge, his Liberation at death is certain.
I am not familiar with the term Paramukta. A Jivamukta attains oneness with Nirguna Brahman - Videhamukti. The term Avatar is a term usually meaning Incarnation of God - Iswara. A freed soul that attains Saguna Brahman attains all powers except creation. A freed soul is not Saguna Brahman. Again in the Brahma Sutras, very last part of the book, 4.4.4-22 discusses this very topic. I will quote only two verses, see the link for my information. Verses 4.4.17-18 says (same link given before):
- (The released soul attains all lordly powers) except the power of creation etc., on account of (Iswara being) the subject-matter (of all texts where creation etc. are described), and (the released souls) not being mentioned (in that connection).
[Sankara's commentary] The question is raised whether those who by worshipping the qualified Brahman attain Brahma-loka and lordly powers, have limited or unlimited powers. The opponent holds that it should be unlimited, because of the scriptural texts, “They can roam, at will in all the worlds” (Chh. 7. 25. 2, 8. 1. 6); “To him all the gods offer worship” (Taitt. 1. 5). This Sutra says that the released souls attain lordly powers without the power of creating, preserving, and destroying the universe. Barring this power they get all other powers. Why ? Because Iswara is the subject-matter of all the texts dealing with creation etc., while the liberated souls arc not mentioned at all in this connection. Moreover, this would lead to many Iswaras, which may give rise to a conflict of wills with respect to creation etc. Therefore the powers of the liberated souls are not absolute but limited, and are dependent on the will of Iswara.
- If it be said (that the released soul attains absolute powers) on account of direct teaching (of the scriptures), (we say) no, for the scriptures declare (that the released soul attains Him) who entrusts the sun etc. (with their offices) and resides in those spheres.
[Sankara's commentary] “He becomes the lord of himself” (Taitt. 1. 6). From the direct teaching of the Sruti the opponent holds that the released soul attains absolute powers. The Sutra says that his powers -depend on the Lord, for the text cited further on says, “He attains the Lord of the mind”, the Lord who abides in spheres like the sun etc. and entrusts the sun etc. with offices. Therefore from this latter part of the text it is clear that the released soul gets its powers from the Lord and depends on Him. Hence its powers are not unlimited.
