you need to check with your particular state if you are getting a passport. I have checked with a website that our state has and it states that either the legal documentation stating custody or a signed release from the non-custodial parent is required. It's always best to call your County Courthouse and check with them to see what you really need before totally sweating about this.
Passports are issued by the United States Department of State. Neither the state nor the county has anything to do with it, nor any kind of a say. The definintive source for all Passport related questions is http://travel.state.gov/passport_services.html. There are new and special requirements promulgated since 7/1/2001 that you must meet to obtain a passport for a minor under age of 14 here http://travel.state.gov/specialreq.html. Please read them carefully. I recommend printing these pages and taking them with you to your local passport issuing office (here it's my post office) so that if you have any trouble, you can point to the regulations.
With respect to a divorce, it is issued by your state courts (possibly at the county courthouse, but state law applies). Whether or not you have authority to take your child out of the country or anywhere else with or without the "permission" of a non-custodial parent is regulated by the terms of your divorce decree.