Sounds like a very complicated setup with some problems (i.e. not up to code). As an amateur I'd definitely recommend an electrician here. This may just be old installation, but from the sound of it, DIY may be how it got this way in the first place.

I don't know anything about arc fault breakers so I'm no help on your original question. But the box sounds like a mess. :-]

GFCI (ground fault circuit interrupters) work by sensing the difference between the outgoing current and return current. A typical fault limit is 5 miliamp difference. I am assuming that you have GFCI also known as GFI. Also, the return current should pass through the same hole in the breaker box as the outgoing current. Trouble may develop if the return path is somewhere else. In fact, anywhere one lead passes through a hole in metal, its partner lead should also pass through the same hole.

My first suspicion is your dryer. If it is a gas dryer, then a likely source for a sneak ground path is the motor or one of the controls. Relay coils can leak. If the motor is a capacitor start type, the start winding and its capacitor is suspect. If the dryer is by electric heaters, there is ample opportunity for heating elements to leak to the frame of the machine.

As a first check, measure the voltage between the appliance frame and a solid ground such as a metal water pipe. Clip on the meter and turn on the appliance while watching the meter. (An analog meter works better than a digital for this test since the voltage jolt may be too short for the digital to respond whereas the needle on an analog will jump or twitch enough for one to see it. However, if short is present and persists for more than a second, a digital meter will also have a response.) There should be no voltage. If you see a voltage, note its value. If it is line voltage, you have a very dangerous situation. Electrocution may result if a user touches a metal part of the machine and a ground such as a water faucet at the same time.

If you are not comfortable with using a volt-ohm meter, then get an electrician or technican who can trouble shoot your setup. The GFI is telling you that something is wrong. Of course, a GFI contains electronics and could fail, but GFIs from major manufactures have a good reliability record. Most manufacturers recommend that a GFI or ordinary circuit breaker should be replaced if it has opened a short.

I would suggest reposting your message in the Electrical Wiring section.