Is Jainism and Nastik (Atheist) Hindu philosophy such as Cārvāka also in the scope of this proposal?
Both Jainism and Cārvāka are part of Hinduism. When I say this, you need a proper definition of Hinduism.
Hindu refers to an identity associated with the philosophical, religious and cultural systems that are indigenous to the Indian subcontinent.
In the Indian constitution, the term Hindu includes Buddhism, Jainism & Sikhism.
P.M. Bakshi: The Constitution of India, p.41:
Article 25 (2)(b) of the Constitution stipulates that the reference to Hindus shall be construed as including a reference to persons professing the Sikh, Jain or Buddhist religion
Thus, all faiths & philosophies which originated in the Indian subcontinent is Hindu.
Even if by definition Jainism can be considered as a sub culture of Hinduism having its origin in India, its tenants and philosophy differ greatly from mainstream Hinduism. When we say Hindu religion we actually mean Vedic religion. So by Hinduism we actually refer to a culture and social order which is based on the teachings and revelations of the Vedas. But Jainism is not based on the Vedas and it has its own sacred texts (Agamas). It has its own separate philosophy and prescribed way for life which vary fairly from Vedic system. It doesn't even accept the existence of a God. So if it is included then we will only be welcoming arguments and disputes like our old Acharyas (scholars) used to have in the past; and I believe it is neither our purpose nor our intention to do so. So Jainism having its own set of doctrines and philosophies should have its separate proposal like Buddhism. It is not a good idea to include Jainism here just to express how huge or great Hinduism is that it encompasses other religions like Jainism. Let's stick to the Vedic schools of thought only for clarity and purpose.
Now regarding Charvak school of thought we can include it. Why? Because it is not a separate doctrine itself having its own set of philosophies and ways for life. It simply rejects the authority of the Vedas and do not have its own existing texts or scriptures. Only 10 to 11 arguments of it are present which criticize the Vedic karmakanda system. These arguments have already been refuted in the past by our Acharyas. I would love to list them here, but I'll save the fun for later. Although there are materialists today who don't believe God, there are hardly any scholars or followers of this particular school of thought. So we won't get much questions or arguments about the Charvak philosophy.
So I would suggest making Jainism another standalone site like Buddism instead of including it here and accepting questions from Charvak philosophy.
Hinduism or Indic religion as is better known is a conglomeration of various philosophies and ideas. It cannot be bound by a single book or a single God. Hinduism, Jainism, Buddhism all have been in the past and present defined very loosely and have shared same boundary. As far as Cārvāka is concerned, it forms an integral part of Hinduism because of the flexible nature of it.
Hence, Jainism and Cārvāka should be included in this topic. At least if not Jainism, Cārvāka must definitely be included.