When I was about 7 years old, I killed a cat for some reason. I don't remember what came over me but I ended up killing it. Can you tell me if I have sinned and what I should do? Is there any special Duaa' I can recite?

All perfect praise be to Allah, The Lord of the Worlds. I testify that there is none worthy of worship except Allah, and that Muhammad is His slave and Messenger.

Since you committed such an action before attaining puberty, you are not blamed. The Prophet said, “The pen is raised for three: one who is sleeping until he wakes up; one who is insane until he becomes sane; and the child until he becomes pubescent.” [Abu Daawood and At-Tirmithi]
Anyway, we advise you to treat animals with kindness and to avoid hurting any animal without a just and genuine reason.
The Prophet said: “Don't take any living creature as a target (for amusement, for training…).” [Muslim]
Allah knows best.