You definitely don't need both the LED and a pilot light, but it could help with trouble shooting. It's just more information.
You show the LED wired correctly.
I like to use a regular outlets because they're cheap, safe, and convenient. That way everything just plugs and unplugs. Simple. All of the ground wires would join with the green wire as you say, ground is shown on a schematic as that upside down triangle of smaller and smaller lines. Their path is often omitted just to simplify the drawing. The ground is really just a redundant neutral wire (the negative, white wire), it provides a second (emergency) path to ground to hopefully stop you from becoming that path (ouch!). That's why you want anything with a hot wire running to it to have a ground wire running to it as well, safety first.
All of your switches, the heating element, and half of the SSR are high voltage (120v AC). We could talk about the current but we're not (the heating element at 12 amps is high current, a pilot light at less than.25 amp is low current). The PID output to the SSR (the input into the SSR), the LED, and the thermocouple leads are all low voltage (~5 volts DC). High voltage could kill you as well as any low voltage equipment it touches. Low voltage will not hurt you but your static electricity could hurt it!
I like to wire the power to the pump from the same box so that the heating element cannot be powered unless the pump is running. This just might save your heating element. Plus its nice to have an on off switch for the pump.
You show the LED wired correctly.
I like to use a regular outlets because they're cheap, safe, and convenient. That way everything just plugs and unplugs. Simple. All of the ground wires would join with the green wire as you say, ground is shown on a schematic as that upside down triangle of smaller and smaller lines. Their path is often omitted just to simplify the drawing. The ground is really just a redundant neutral wire (the negative, white wire), it provides a second (emergency) path to ground to hopefully stop you from becoming that path (ouch!). That's why you want anything with a hot wire running to it to have a ground wire running to it as well, safety first.
All of your switches, the heating element, and half of the SSR are high voltage (120v AC). We could talk about the current but we're not (the heating element at 12 amps is high current, a pilot light at less than.25 amp is low current). The PID output to the SSR (the input into the SSR), the LED, and the thermocouple leads are all low voltage (~5 volts DC). High voltage could kill you as well as any low voltage equipment it touches. Low voltage will not hurt you but your static electricity could hurt it!
I like to wire the power to the pump from the same box so that the heating element cannot be powered unless the pump is running. This just might save your heating element. Plus its nice to have an on off switch for the pump.