I have heard that if you have a non-custodial child and do not communicate with them for 6 months, the custodial parent can take the child legally away from you. Is this accurate?

If both parties no longer reside in the county the case was opened in, one in a different county, one in a differet state, would the case be automatically transferred to where the custodial parent lives? What if the custodial parent wants to close the case? Would that parent have the option?

It all depends on where in the world you live. Since you did not provide your location we cannot answer you properly.

The state is Michigan

While the law may have guidelines that allow a court to terminate rights, as a practical matter this very rarely happens. Courts are very reluctant to issue a TPR and ONLY a court can.

As for jurisdiction, if both parties have left the jurisdiction then it goes to where the child lives.