Is caste based on birth? Is it inherited? Do any scriptures support birth based caste system?
This is discussed in detail in Mahabharata.Vana Parva.Tirtha-yatra Parva. There is a conversation between Nahusha (in form of serpant), fifth in descent from the Moon and Yudhishthira:
The serpent said, 'O Yudhishthira, say--Who is a Brahmana and what should be known? By thy speech I infer thee to be highly intelligent.'
To answer this,
Yudhishthira said, 'O foremost of serpents, he, it is asserted by the wise, in whom are seen truth, charity, forgiveness, good conduct, benevolence, observance of the rites of his order and mercy is a Brahmana. And, O serpent, that which should be known is even the supreme Brahma, in which is neither happiness nor misery--and attaining which beings are not affected with misery; what is thy opinion?
Then, the serpent (Nahusha) objected this:
The serpent said, 'O Yudhishthira, truth, charity, forgiveness, benevolence, benignity, kindness and the Veda 1 which worketh the benefit of the four orders, which is the authority in matters of religion and which is true, are seen even in the Sudra. As regards the object to be known and which thou allegest is without both happiness and misery, I do not see any such that is devoid of these.'
The Yudhishthira replied
Yudhishthira said, Those characteristics that are present in a Sudra, do not exist in a Brahmana; nor do those that are in a Brahmana exist in a Sudra. And a Sudra is not a Sudra by birth alone--nor a Brahmana is Brahmana by birth alone. He, it is said by the wise, in whom are seen those virtues is a Brahmana. And people term him a Sudra in whom those qualities do not exist, even though he be a Brahmana by birth. And again, as for thy assertion that the object to be known (as asserted by me) doth not exist, because nothing exists that is devoid of both (happiness and misery), such indeed is the opinion, O serpent, that nothing exists that is without (them) both. But as in cold, heat doth not exist, nor in heat, cold, so there cannot exist an object in which both (happiness and misery) cannot exist?"
The serpent (Nahusha) again objected:
The serpent said, 'O king, if thou recognise him as a Brahmana by characteristics, then, O long-lived one, the distinction of caste becometh futile as long as conduct doth not come into play.'
Yudhishthira then, replied
Yudhishthira said, 'In human society, O mighty and highly intelligent serpent, it is difficult to ascertain one's caste, because of promiscuous intercourse among the four orders. This is my opinion. Men belonging to all orders (promiscuously) beget offspring upon women of all the orders. And of men, speech, sexual intercourse, birth and death are common. And to this the Rishis have borne testimony by using as the beginning of a sacrifice such expressions as--of what caste so ever we may be, we celebrate the sacrifice. Therefore, those that are wise have asserted that character is the chief essential requisite. The natal ceremony of a person is performed before division of the umbilical cord. His mother then acts as its Savitri and his father officiates as priest. He is considered as a Sudra as long as he is not initiated in the Vedas. Doubts having arisen on this point, O prince; of serpents, Swayambhuba Manu has declared, that the mixed castes are to be regarded as better than the (other) classes, if having gone through the ceremonies of purification, the latter do not conform to the rules of good conduct, O excellent snake! Whosoever now conforms to the rules of pure and virtuous conduct, him have I, ere now, designated as a Brahmana.'
With this answer the serpent (Nahusha) agreed and freed Yudhishthira' s brother Bhima:
The serpent replied, 'O Yudhishthira, thou art acquainted with all that is fit to be known and having listened to thy words, how can I (now) eat up thy brother Vrikodara!"
Are children of Brahmin parents also Brahmin by caste?
It is not necessary .. It is not that simple as well.
Anyone is a Sudra without Samskaras (initiation) and so is a Brahmin even though he is one by virtue of birth.
Manu Smriti 2.172. (He who has not been initiated) should not pronounce (any) Vedic text excepting (those required for) the performance of funeral rites, since he is on a level with a Sudra before his birth from the Veda.
So, what will be the castes of the sons of such a Vratya Brahmin (who has not received the Samskaras)? certainly not Brahmins , they will be Sudras (Vratyas) only.
16 Sons fathered by an uninitiated man are Vra¯tyas, excluded from initiation––so state the wise with equal reference to the three classes.
Baudhayana Dharma Sutras 1.16.10
And, Samskaras are as many as 40 in number, starting from the Garbhadana, as stated in Gautama Smriti with Upanayana (initiation) being just one of the them.
And, to maintain the purity of caste system, as is mentioned in the Smritis, each of those Samskaras are to be performed in their prescribed times, generations after generations.
Assume that that has been done but the discussion does not end there.
Because, anyone can loose his caste by performing certain actions in his lifetime. And, it is quite easy for a Brahmin to loose his caste as well.
Manu Smriti 10.92. By (selling) flesh, salt, and lac a Brahmana at once becomes an outcast; by selling milk he becomes (equal to) a Sudra in three days.
In this case, what will you call the sons of this Brahmin who has become an outcast or a Sudra?
And, a Brahmin of bad conduct is also a Sudra or a Chandala. What will his sons be?
He who sets his face against righteousness, is but a Chandala by his acts. He gains nothing by being a mendicant, or by worshipping the household fire. (A Parashara Smriti verse)
Therefore, birth is only the first (and one) determining factor of caste, Samskaras and conduct being the other and more important factors.
Yes, caste is based on birth. There is scriptural support for it.
Manusmriti (10.5).- In all castes (varna) those (children) only which are begotten in the direct order on wedded wives, equal (in caste and married as) virgins, are to be considered as belonging to the same caste (as their fathers)
Āpastamba (2.13.1).—‘Sons begotten by a man who approaches in the proper season a woman of equal caste, who has not belonged to another man, and who has been married legally, have a right to follow the occupations of their castes.’
Viṣṇu (16.1).—‘On women equal in caste to their husbands, sons are begotten who are equal in caste to their fathers.’
Yājñavalkya (1.90).—‘From women of the same caste as their husbands are born sons of the same caste.’
Baudhāyana (1.17.2).—‘Sons of equal caste spring from women of equal caste.’
The genetic makeup of the 4 castes are different. Brahmanas have Nadis in their body situated in such a way as to receive the vision of the Lord. Kshatriyas are genetically strong and physically fit. Vaishyas have a mind and body conducive for conducting business. Shudras are naturally pessimistic, lazy, simple-minded, and materialistic. That's how their bodies were constructed.
Source for Nadis in a Brahmana's body situated in a particular way. At 39:00 minutes in the video.