Karn and Arjun

To know about Karn's character
See Kevin McGrath's "Karna as Sanskrit Epic Hero" (Brill, Leiden) and 
R. Cecily-Katz: "Arjuna in the Mahabharata" (Motilal Banarsidass) . 

Pradip Bhattacharya's review of Sawant's "Mrityunjaya" on www.boloji.com analyses Karn's character and dispels many popular misconceptions.

Is there any reference to rivalry between Soorya and Indra as a possible motivation or justification for the enmity between Karn and Arjun?"   
Not really as motivation or justification, but it depends partly how you look at it, and the poets do certainly play with the symbolism in this connection, especially in the final battle between the two. Yaroslav Vassilkov discusses sun/rain imagery in that battle in his (possibly rather obscure) 2001 article "The Mah Œbh Œrata similes and their significance for comparative epic studies" (Rocznik Orientalistyczny vol. 54 no. 1, pp. 13-31).
My Note - MBH, Aadi/75 says that Kashyap had 12 Aaditya from his eldest wife, Indra being at their head, and aso Vivaswat (Soorya) - two heads cannot live together, maybe that is why this rivalry has grown.
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I have pointed out in my book on the BR Chopra TV MBH serial that the Indra-Soorya opposition paralleled by Arjun-Karn is prefigured in the Raamaayan where these same 2 gods are fathers of the rival brothers Sugreev and Baali. Note that in both cases it is Vishnu-Raam/ Vishnu-Krishn with whose aid Indra-Sugreev/ Indra-Arjun kills Surya-Baalee/Soorya-Karn. The Rig Ved has evidence of an Indra-Soorya confrontation, the former wrecking the latter's chariot wheel. the only difference I can note is that Soorya is the last of the Aaditya (Maartand) while Indra is the first; on the other hand, Karn and Baali are the eldest, not Arjun and Sugreev.
[Pradip Bhattacharya]
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"Mrityunjaya" may be one of the best sources of information about Karna. Any English translation available?
English translation of Mrityunjaya by P. Lal and Nandini Nopany published by Writers Workshop, is available from profsky@cal.in. A critique of this by me is on www.boloji.com, "I am myself alone". If you look at the annotated bibliography on the resources page, many of your queries may get answered.
[Pradip Bhattacharya]
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Kevin's book on Karn 
is available on the website www.brill.nl  The book is available online free of cost.
In Brill's page they give a pointer to Google books site. This Google is trying to become a vast book library, these days. In Google search goto > more > books > and in this book search category, there are two options, full view books, or all books. select full view.
OR, in Brill's page - read book or read through Google. besides Kevin''s book there are a host of other books too in full view > full screen. Of course, saving is disabled.