Can the court cutoff a parent's rights to a kid if the divorce is on appeal?

If you have an attorney who is handling your case you should advise him or her of your concerns. If you do not have an attorney you should consider hiring one to assist you with what sounds like a complicated matter. A court will will not "cutoff" someone's parental rights based solely on a failure to pay support. Best of luck to you!

All of the trial court's issues are suspended until the final ruling by the appellate court.

It depends, everything contained in the divorce decree is off limits to address while the appeal is on going, however, if there were emergency circumstances (unable to determine based off the facts- and child support being unpaid doesn't qualify) then you may be able to temporarily cease her ability to possess the children. I would advise that you speak with an attorney in order to guide you once fully apprised of the facts of the case.