Electronic – IEC socket rated for 250V at 10A – can it be used for 13A @230v

The Phidget board that you refer to has digital outputs only.

The voltage-controlled dimmer has an optically-isolated analog input, which takes 0-10V on the non-mains side and generates an analog signal on the hot-side (the side connected to the mains) which is read by the dimmer board micro and used to control the dimmer.

The main point is that your USB board (and whatever circuitry you intend to use to control the optoisolator current) must only be connected to the photodiode (non-hot side of the optocoupler). If you try and interface it directly to the dimmer micro, plug the circuit into the mains then plug a computer into the USB port, you're going to have a catastrophic failure on your hands (since you'll be introducing an earth connection to the primary circuitry) or worse, end up electrocuting someone.

You must make sure that inside your plastic box, no part of your Phidget USB board is located near the mains areas of the dimmer board. In North America, depending on the voltages involved and the exact standard, 4 to 8mm spacing is fairly common.

Use wires with 600V insulation rating to connect between the Phidget and the optoisolator, in case they happen to come near any of the mains circuitry along their travels.

Using the relay to disconnect the neutral line doesn't really sound like you know what you're doing. You're exactly wrong in the safety department. In a perfect world, it's the hot that you want to switch. With only neutral switched, the heating coil is always energized, waiting to shock you.

Additionally, relying on a polarized electrical cord to try to make sure it's the hot you're switching has its own problems. You're better off using a double pole relay to fully disconnect the toaster from the electrical supply. Use a relay that is designed specifically to switch both legs of a power supply.

Also don't need to heat-shrink, glue, or electrical tape the wires, that's overkill and amateur hour. Hiding things in an attempt to increase safety is a sure way to decrease safety.

Use the project box, and mount your circuitry (including relay) on perf board. Simply solder the relay inline on the hot wire. Keep your low voltage relay control wiring (and arduino circuitry) physically separate from the relay and line voltage power wires. Do not let low voltage and high voltage come near each other, except at the relay, and even then, those come from different directions. Physically secure the power wires so they can't accidentally be pulled out. Be neat.

Your heating coils should be shielded from your pcb by a heat spreader, and your thermocouple should be next to the pcb, and not be touching anything. It's the air that heats up the pcb, and the air that heats up the thermocouple. You trying to bake the pcb, not broil it.

IMPORTANT SAFETY NOTE:
The electrical power that comes out of your outlets is extremely dangerous. It's not the fun-times easygoing stuff of 15 volts and below. It will surprise you, and given the opportunity, it will kill you.

From your question, you really don't seem to know what you're doing. I'm not trying to insult you, I just don't want you to die. Advice from the internet is not going to make this any safer. Sit down with someone in the real world who has worked PROFESSIONALLY with household electrical circuits and have them look over what you are doing.

Please be safe.