Once the economic and social usefulness of the motor car was demonstra

Hi akijuneja.
I'm glad to help.

Once the economic and social usefulness of the motor car was demonstrated and with its superiority to the horse being proved, much of the early hostility to it in rural regions disappeared.

(A) and with its superiority to the horse being
Wrong.
-" and with" does not make sense.
- "being" is not main verb, thus the clause "with its superiority to the horse being proved" is not complete and fragmented.

(B) and its superiority over the horse had been
Wrong.
- "superiority over" is not correct idiom. The correct one is "superiority to"
- Past perfect "had been" is not necessary.

(C) and its superiority to the horse
Correct.
- "and" is parallel marker. The two clauses before and after "and" are parallel.
- "superiority TO" is correct idiom.
- "Prove" is intransitive verb which does not require any further elements to make its meaning complete. Thus, we don't need passive voice "was proved" here.
For example: the new drug proved effective

(D) its superiority over the horse
Wrong.
- "superiority over" is not correct idiom. The correct one is "superiority to"
- we need "and" to separate two clauses. Without "and" the sentence is nonsensical.

(E) with its superiority to the horse having been
Wrong.
- "with" is not correct. The correct conjunction word is "and".
- "having been proved" is WRONG grammar. (If you think about present perfect continuous in passive voice, the correct form is: have/has + being + Past participle)

Hope it helps.