Running two separate cables for mic/headphone is going to be bulky unless you intend to strip the jackets on both cables and sleeve everything into a single cable. You didn't list sleeving in your bill of materials, so I'm going to assume that isn't the plan.
Instead, you can get away with using just the quad cable if you're willing to share the ground wire between both of the drivers and the mic. Use one wire as the signal for the left driver, another wire as the signal for the right driver, another wire as the signal from the mic, and the last wire is ground shared between all three signals. You'll have to splice the ground wire into three separate wires inside the headphone cup in order to connect everything and also at the end of the cable to run the ground wire to both jacks. This is how I wired my AD700 mic mod and it works fine, some noise is introduced when I connect the mic but it isn't noticeable while gaming. I just disconnect the mic when listening to music.
Alternatively, you can use the shield as a 5th wire and use that as ground for the mic if you wanted to keep it separate. Canare braids their shield, so I don't know how easy or not it is to twist into another wire. Mogami, on the other hand, simply twists theirs and it is quite easy to make into a fifth wire.
As for the actual wiring and soldering job, a multimeter will be handy. Most of them have a mode to test for continuity, you just touch the two probes at the ends of what you think is the same wire and it'll beep to confirm it for you. It is extremely important to make sure you get all the signals correct. When wiring the headphone jack, the tip is left signal, the ring (between the tip and the sleeve) is right signal, and the sleeve is ground. For the mic jack, you're going to have the wire the signal from the mic to both the tip and the ring and ground to the sleeve, I've worked with that particular mic and that is how it is wired by default.