Peasants who were Catholics, taught by Catholic doctrine and engaged in a Crusade started by the Catholic Church. Yet you don't see such mobs systematically destroying entire Jewish villages in Protestant areas, and you don't see it in Russian Orthodox areas until the 1500s.

Protestantism didn't even exist until the 1500s. I don't see why/how you're making the comparison.

So the Protestants had 500 years to do so and they didn't do it. But the really relevant group above is the Russian Orthodox who didn't start heavy persecution of Jews until later, and did so generally after absorbing memes from Catholics.

I was under the impression that most intelligent people at least knew that the Church had killed all the Cathars, and that's why Cathars don't exist anymore.

While most people who know about Cathars probably know that they were wiped out by the Catholic Church, you are likely overestimating how many people know about them at all.

Catholics are responsible for universities.

Actually, similar institutions existed elsewhere, especially in the Islamic world. The House of Wisdom functioned in many ways very similar to a university. By many accounts, Al-Karouine is the oldest university in existence, and clearly predates the European universities. There's a serious argument that Islamic schools played a vital role in influencing the establishment of universities in Christendom (although I think the influence here is probably often overstated).

Well, they shouldn't do it anymore, for obvious reasons.

What obvious reasons? Question: if you somehow time traveled back to 1491 and the pope asked you whether he should endorse a new Inquisition in Spain, would you tell him yes?

Responding to your ETA remark, I think you really don't get the problem. Aside from the potential problem of projection. Essentially you made an extremely positive remark about the Catholic Church and then aren't apparently seriously defending it while claiming that criticism is somehow a "fight' between which "side" is better. But that's not the issue here. The issue here is that you said that the Catholic Church had an "absurdly good" track record. You didn't say that it was better than some others or that on balance it has done more good than harm (both of which are not unreasonable claims), but rather you made a much stronger claim. And that claim simply doesn't hold ground. The Catholic Church like most other religions and long-term institutions is a mixed bag. There's been good and there's been bad. They've helped preserve learning and they've burned books, they've saved lives and they've taken them. That's not "absurdly good" by any reasonable notion of that term.