“Does killing animals usually indicate psychopathic tendencies in children?”

In over 90% of the cases, yes, it does. It all depends on the child's upbringing and how the animal killing is viewed.

Killing of domestic animals is actually unlawful and a reportable offense.

Not necessarily. Many children grow up on farms or where having a few chickens or other animals that are raised for food is totally normal. Others grow up in families in which hunting is totally normal. What is of concern is when the child seems to enjoy cruelty, deliberately causing suffering, or seems to get a charge out of pointless killing. We also look at how they relate to their fellow humans. If we see empathy, consideration, rule-following, respect, and kindness, and we don't see impulsive risk-taking, exploitation, disregard for the rights or feelings of others, and unusual selfishness, then there is less cause for concern.
I hope this helps.

Killing intentionally? If so, that is reason to get them to see a behavioral specialist and possibly a psychiatrist ASAP. It is not normal behavior and may indicate some underlying psychosis and/or rage.

See this link for some insight into your question:
https://www.psychologytoday.com/us/blog/in-excess/201611/the-psychology-animal-torture