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.
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Looks like that subjunctive mood.

Choose (C) and click confirm.

C looks good,

but i think C should have a "was", because the first clause is passive "was demostrated" so for ||ism second clause should also have "was proved"....

its superiority "over" the horse...

Would go with C

ha ha ha ha, that's reality! (still trying to laught )
Your avatar looks cool

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 - incorrectly and unnecessarily placed
(B) and its superiority over the horse had been - wrong idiomatic usage
(C) and its superiority to the horse - Correct choice
(D) its superiority over the horse - wrong idiomatic usage
(E) with its superiority to the horse having been - "with" incorrectly used and "having been" unparallel usage of tense

Split 1: Should there be an "and"

Compared to the other options with and its, which make it sound as a run on sentence, there needs to be an AND. Elimnate D and E on this basis.

A v/s B v/s C

The correct idiomatic expression is superior to. But another approach for eliminating A and B are as follows:

A uses the word being, which gives the statement a passive tone.
1. GMAT doesn't prefer passive.
2. The entire statement should be in the same tone

B uses had been, a past perfect. This is not the case. Past perfect is used when an action was completed before another action completed in the past.

Hence, C.

Please help me in evaluating this approach
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(A) and with its superiority to the horse being
(B) and its superiority over the horse had been
(C) and its superiority to the horse
(D) its superiority over the horse -> lack of "and" -> a run-on sentence.
(E) with its superiority to the horse having been