Converting to Hinduism and Caste
There has already been a lot of discussion on converting to Hinduism and about caste system here on this site. The OP's question is slightly redundant. But I will answer the last part which hasn't been talked about here.
And, what practices for worship do the converts follow? If they follow a certain sect's way, do they automatically get considered one of that sect?
A sect is not a Jati or caste. For example, Vaishnavam or Shaivam is a sect and it has followers from many Jatis.
A brief note on Caste :
Caste as we know of today is corruption of the Varna-Jati-Kula system. And this corruption was done by European Indologists' inability to comprehend the system.
Varna was occupational classification. Jati was occupational sub-classification and Kula was lineage. One was born into a Varna & Jati but was free to switch his Varna based on his/her aptitude & abilities. At times a whole Jati switched Varnas even till as recent as 1930s1.
Caste of converts to Hinduism:
When non-Hindus who become Hindus, they don't have to be part of any Varna or Jati unless they choose to, by taking up initiation. (Note that unlike some other religions, becoming a follower of Hinduism itself does not need initiation. One is automatically a Hindu when he/she follows Dharma. Refer to any answer to another question here).
For example one can follow the Vaishnava sect(like ISKCON) and not belong to a particular Varna/Jati. But if one decides to pursue the creation and distribution of Vedic knowledge under the sect full-time, they can choose to be initiated as a Brahamana(who by definition does only that). In fact ISKCON has a written test which needs to be cleared to become a Brahmana2.
For other traditional sects too, if one is qualified, he can take up initiation to be an acharya/brahmana. For example an archarya of a Srivaishanava mutt in Australia who theoratically is a Mleccha(foreigner) has been initiated as an Iyengar Brahmin by the Jiyar of Sriperumbudur3.
Hence if one merely wants to follow Sanathana Dharma but not dedicate his/her full time to a particular profession, they need not be initiated into a Varna or Jati but just remain a follower of a sect.
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