Do DIY headphone cables need to be braided?

If you use the Mogami as-is (with its rubber sheathing on), it will be more than properly shielded. Under the sheathing is a layer of wiring that is there to provide shielding. And beneath that, the individual wires are twisted, which purports to cut down on interference as well.

Translation: You don't have to braid, although it is easy once you get a system/rig set up. But you will need to learn to solder. Look online for videos and get practicing before you attempt on your headphones.

I've recabled headphones and built cables with Mogami as-is and I've braided individual strands for aesthetics as well. All sound great.

Headphones are low impedance, which means they are essentially immune to electro-magnetic interference. And unless you have a balanced amplifier and a balanced cable with your he-400, you're using a 3-wire cable where one of the wire is essentially grounded. This creates an asymmetrical connection where there is no common-mode rejection. Only balanced cables, connected to properly designed balanced gear, have common-mode rejection

Shielding an headphone cable is probably overkill if you're gonna use them at home. Though there are environments where the shield of a 3-wire cable, connected to ground, does help. The reasons why star quad cables are shielded is because (1) they were designed for microphones and interconnect, which have high impedance inputs and (2) they are meant to be used on stages and studios saturated with EMI. The shield is not needed for headphones. In fact, many people who used those cables for headphones simply removed the sheath and the shield, and used only the twisted quad.

Twisting wires together is more effective than braiding. All manufactured balanced audio cables are twisted wires. Braiding is just a pretty alternative invented by DIY'ers to build their own cables. It's a bit more convenient if you're not gonna sheath the cable. It allows for a looser braid, which is more flexible, without breaking apart.

Braiding is mostly for aesthetics and to remove mechanical noise from the wires bumping into each other. If you're going for the best bang for the buck, just get mogami or canare starquad which costs about a dollar per foot. Though I don't really like to use those cables for headphones since they weigh too much and doesn't have the wow factor of a custom made cable.