That's an excellent question that I don't know the answer to. However we definitely studied the "What's Opera Doc" and "The Rabbit of Seville" in my music history classes in university, talking about how the influence of opera came into the public lexicon.

That being said, the lack of interest in classical music wasn't really a problem in the 50s and 60s the way it is now. Music classes were still very much part of the public school curriculum, and kids taking piano/violin lessons was still very much the norm. If anything, the music of Wagner and Rossini would have been far more familiar to a 1950s audience than it would be to one of today.

There was a post a while back that listed the Met productions in the late 40s-early 50s, and it was absolutely PACKED full of Wagner. Far more Wagner that you would see in any opera house today (other than those that specialize in it, like Bayreuth).

The cartoons you are thinking of, namely The Rabbit of Seville and What's Opera, Doc? weren't in heavy rotation on television during the fifties, sixties and seventies. Of the thousand-plus Looney Tunes and Merrie Melodies cartoons, only a handful featured classical music. You were far more likely to hear covers of popular songs of the day or the high-energy arrangements by Carl Stallings of Raymond Scott et al.

Originally Associated Artists Productions bought television rights to all of the Warner Brothers film catalog prior to 1950. This included all Looney Tunes and Merrie Melodies cartoons up to 1948. These were licensed to independent television stations and were played frequently on weekday, daytime television for children.

Warner Brothers kept the television rights to cartoons after 1948 and showed them first on prime time and then on Saturday mornings on one of the major broadcast networks (ABC, CBS, NBC). So, children watching, for example, The Bugs Bunny Show in the late sixties wouldn't see either of the two aforementioned classics more than once a week. From my experience, I don't ever remember seeing either during Saturday morning programming. I first saw them in the seventies when they aired during prime time as a television special.

So, this is kind of a long introduction but I wanted to establish that classical-music themed cartoons in general, and the one you mentioned in particular were rarely seen on daytime, children's television so I think their influence on the musical tastes of children was minimal.