Is the 4 wire outlet for a dryer, stove, or something else? How many amps is the receptacle? If it's for a dryer or stove (they are different) buy the appropriate whip at a big box store. You wire the black and white wires from the compressor to the hot terminals which are the vertical blades of the plug. The green wire connects to the plug ground. Leave the neutral of the plug unattached. Make sure the double pole breaker is sized correctly for the cable (#10 is 30A, #8 is 40A). The breaker protects the cable, not the compressor.

There are only two ways to connect wires and devices in a circuit, parallel and series. When you connect any circuit wires in parallel to a fuse, switch, circuit breaker, or GFCI , that device will not have any effect on those circuits. Perhaps you can use some lern'in? TEXTBOOK "A fuse, (circuit breaker, switch, or GFCI) is connected in series with the power source so that the entire current passes through it." "If two LOADS are in series, then removing one load will disable the other." A circuit doesn't have to have two LOADS to have things connected in series. Two wires can be connected in series. You would need at least 3 wires to even connect them in parallel. "A typical example is the type of christmas string lights where you remove one bulb (ie. load) and all the other ones go out." A typical example is a switch that controls a string of lights. When you turn the switch off, all the bulbs go out. Another example is a GFCI. When the GFCI trips, all the bulbs go out.

"Also, where is my ground? " I was just about to ask that. From your picture, it suggest that the wire could be run in conduit? ( the usual reason that the black wire would just run through) If it IS in EMT, then everything is (should be) grounded by the conduit, but you either need a ground pigtail for the receptacles, or a special grounding receptacle. If it is NOT in conduit, then; CAUTION: The permission to replace nongrounding-type receptacles with GFCI-protected grounding-type receptacles doesn't apply to new receptacle outlets that extend from an existing ungrounded outlet box. Once you add a receptacle outlet (branch-circuit extension), the receptacle must be of the grounding (bonding) type and it must have its grounding terminal grounded (bonded) to an effective ground-fault current path in accordance with 250.130(C). Chicago has some different codes on this such as requiring conduit, and some states, such as Massachusetts, require you to start any new wiring from the properly grounded panel. [no circuit extensions allowed on Romex without a ground, or grounding by 250.130(c)] 210.7 (B) Multiple Branch Circuits. Where two or more branch circuits supply devices or equipment on the same yoke, a means to simultaneously disconnect the ungrounded conductors supplying those devices shall be provided at the point at which the branch circuits originate. HOWEVER, since the circuits need to be on an AFCI breaker, you should in most cases need to use a 2-pole breaker. There are some exceptions.

I thought I replied to this post earlier today, but I guess it did not go through. The wiring for the receptacle and the light beyond it sound similar to what I have run across a couple of times in my house. If the power is running to the light switch, then they may have run 14/3 from the switch to the 1st outlet and used the white/black pair to power the receptacle, and the white/red pair passes through the receptacle to provide power to the light that is beyond. The only odd part is that usually you only need 14/2 from the receptacle to the light to make this work, but you said they had two reds nutted together in the box. You can use a similar scheme in reverse if you have power at the light, then 14/3 from the light to the receptacle, and then 14/2 running to the switch. The red wire is just used as a traveler that passes through the receptacle to provide a switch loop from the light switch to the light. I was very confused the first time I found this type of wiring scheme, since it looked like I was working with a 3 way switch circuit. However, it seemed that it was just a way to save running one more 12/2 cable from the light switch to the light. Bruce.