Incarnations are rare indeed and their case is different.

As for the rest, God does not discriminate between any of His creations. However, the divine experiences and visions depend on the extent of Sadhanas (spiritual practices) that one has performed. The sadhana that one performs is a continuation of the sadhanas of his previous births.

One can no doubt, have divine visions and blissful experiences in this birth itself, if he incessantly chants the Maha Mantra with complete faith and devotion.
If Adi sankara Bhagavadpada had been initiated into Kriya Yoga by Babaji, his disciples like Thotakachara or Hastamalaka etc would have mentioned it in their works. Sri Adi sankara Bhagavadpada himself has dedicated all his works to his guru Sri Govinda Bhagavadpada. He would have mentioned his initiation by Babaji in his books, had that been the case. This being a case which cannot be concluded without any trace of doubt, we cannot say any thing further conclusively on initiation of Sri Adi Sankara’s initiation by Babaji.
Yes. What Swami Vivekananda says is absolutely right. You will need to come out of the cocoon called Samsara, to eventually get liberated. Now the question is, how do you burst your own cocoon and come out of it? The easiest and simplest way to burst open the cocoon is by chanting God’s Divine Name.
Sri Raghavendra was a great Mahan and a great Hari Bhakta. He is considered an incarnation of Prahalada Swami, who is the supreme among Bhagavatas. Moreover, the term ‘Raghava’ in the name of Sri Raghavendra denotes Sri Rama. Maha Mantra Kirtan is nothing but Hari Nama Kirtan,which is the favorite of the Bhagavatas. Hence chanting the Maha Mantra will indeed please Sri Raghavendra Swami. He will definitely accept your Kirtan you chant the Maha Mantra in front of his picture.
The difference between the view of J Krishnamurti and an atheist can be likened to that between opposite poles.

The atheist or non-believer denies everything that is about God and spirituality such as the existence of God, authenticity of scriptures, spiritual ideals, role of the Guru etc from a very superficial viewpoint. There is no honest, unbiased, open-minded probe into the deeper levels or a genuine yearning to find the truth. It is more out of hatred rather than open-mindedness.

The rational approach that J Krishnamurti advocates is one that is based on a genuine thirst to know and experience the ultimate truth. J Krishnamurti says that one should find the truth all by himself and not accept it just because it is written in the scriptures or it was told by the Guru. He says that if we simply accept it making no effort to progress it is a reflection of our laziness. It takes us nowhere. That is the context in which JK denies the word of scriptures, Guru or any set of predetermined principles. It is not due to hatred but due an absolutely unbiased, open-minded, sincere, focused approach to find the truth without any presumptions. This path too, if followed in the right way, will lead to liberation because that is the Ultimate Truth.

Besides, we can view JK in this way too.

God, out of His infinite mercy towards all beings, sends Mahatmas and saints to show them the way. The world is full of all kinds of people with varying interests, attitudes and preferences. Therefore Mahatmas too, vary their approach in order to show the Truth to people based on their varying attitudes. Sri Ramakrishna came for a particular purpose and Bhagavan Ramana came for another. Likewise, JK came to show the way for those people who were not oriented towards classical Sanatana dharma and had lost faith in such things