I just want to know what makes a Brahmin. Does that depends on the karma? Does that depends on the fate? Does that depends on learning Vedas?
Some say that next birth of Brahmin is a cow. Is that true?
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The qualities of Brahmana are as followed:
BG 18.42 - Steadiness, control, austerity, cleanliness, tolerance, sincerity, knowledge, science and faith are certainly Brahma(na)'s natural Karma
Anyone with predominantly possessing above qualities is Brahmana.
The varna-s are based on natural qualities or karma. Our natural qualities are mostly inherited from our forefathers, hence family lineage (viz caste) also has some role to play, even in today's time. Ultimately in which lineage a person would born is fate/luck. :-)
More details of what various professions may belong to 4 varna-s are described in this answer.