No. Natural is: Her first performance was Adele’s song, “Someone Like You.”

For question one, the sentence sounds strange to me. Using "on" here makes it sound like the person performing is not the singer. For example, a pianist would be ON the song. The singer would be giving a performance OF the song. If "she" is the singer, then "she had the first performance" would be incorrect. You can say she "gave" the first performance, that is best. However, there is another problem. Your sentence could make it sound like there was more than one performance of the same song if we change it (as we need to) to "of." So I would suggest: She gave the first performance, a cover of "Someone like You," originally sung by Adele. Or She gave the first performance, which was a rendition of "Someone like You," originally sung by Adele. (slightly more formal) Hope that helps!