Is it true that Hawa's jealousy upon seeing herself in the water led to women's jealousy, and that Qabil's killing of Habil was the reason for the spread of killing on Earth?

There is no mention of anything that proves Eve's jealousy. Cain was the first to institute murder, and he bears the sin of everyone who followed him. There is no evidence that if he had not killed Abel, murder would not have occurred thereafter. One should seek forgiveness for the believers who came before and direct efforts towards acquiring beneficial knowledge and guiding those who are astray.