Both the parent and the child in the dream symbolize parts of yourself. It seems to be showing you that your "adult," responsible, rule-keeping, cautious self is "killing" the part of you that is playful, curious, adventurous and fun. Note that dreams always exaggerate and over-dramatise images, so the killing should not be taken literally, but more in the sense of a "kill-joy."
This seems to be a typical dream that expresses parental concern for the child. It could refer to some influence or person the parent fears might be harmful.
it means that you should give your child more freedom because you're probably an uptight parent.
The killing of a parent or close relative.
Unless you father is terminally ill or his life is at risk for some other reason, this dream is not forecasting the future. Dreams of a parent dying are not uncommon and reflect a young person's changing relationship with that parent. The former relationship of dependent child to protective parent is gone, and that absence can be illustrated in a dream as death. The dream means that you cannot go back to the old parent/child relationship, but must now begin a new way of relating to your father as increasingly equal adults and friends.
it can mean that you sense danger for you child
Dream child or little dream
If the dreamer is the parent of a young child, the dream could express natural parental fears and worry for the welfare of the child. It need not predict any future event. Alternatively, the "child" might symbolize a project, assignment or idea that is important to the dreamer and in danger of being canceled or otherwise threatened.
No it does not mean that you are killing an innocent child, it just prevents an unwanted pregnancy. An abortion means you are killing an innocent child. Birth control just means your being safe.
that you are a cat person
The question cannot be answered without further information; a dream about killing domestic animals for food or killing dangerous predators would be very different from a dream about the cruel killing of pets, or senselessly shooting harmless wildlife.
nothing
It doesn't mean much. Most likely you had the dream when you were little or visioned the creature when you were younger. So yeah, it doesn't mean anything.