Let's analyse each answer choice:
(A)and with its superiority to chloroform being
The part: '...and with it's its superiority ... ' is sounding awkward and ungrammatical.
Also, "the physiological effects and anesthetic qualities of diethyl ether were demonstrated" is in the passive past tense, while "with its superiority to chloroform being proved by William Morton" is in the present progressive. So the parallel structure is not maintained.
Also, I think the usage of 'being' is generally wrong, when it comes to the GMAT, although there is no definite rule for that.
(eliminate)
(B) "and its superiority to chloroform"
This seems to be changing the intended meaning of the sentence. Also, I think that there is a missing verb in this. So the sentence is sounding un-grammatical.
(eliminate)
(C) "and its superiority over chloroform had been"
This maintains parallel structure with the previous clause and uses the correct preposition "over" to compare diethyl ether and chloroform.
The usage of had been (past perfect) denotes th e correct sequence of events. It suggests that William first proved something and then the medical community adopted that.
(let's keep it for now)
(D) "its superiority over chloroform"
This option lacks the conjunction "and", and the sentence sounds grammatically incorrect and awkward.
(eliminate)
(E) "with its superiority to chloroform having been"
The usage of "having been" is awkward and unnecessary.
(eliminate)
I will go with option C.