Hey Moggy,
since they have to be flexible, you need stranded wire. If you want to use the same connectors as on the stock cables, the crimp contacts are for 24 AWG – 18 AWG, so I would take 18 AWG. If you want to make a cable identical to the stock cable, which has neither PE wire nor shield, you simply need a 4-wire cable.
Be sure to have the correct crimp pliers ready as specified here.
Of course in engineering, wires are not measured in wire diameter but in cross-sectional area, because the cross section is directly proportional to its strength and weight, and inversely proportional to its resistance. The cross-sectional area is also related to the maximum current that a wire can carry safely. Wire diameter is not. Wire sizes are standardized internationally in IEC 60228.
AWG is issued by ASTM and is mainly used in the US. Molex and Amphenol specify wire strength in AWG units instead of IEC 60228 
, so what can you do.
Here is a nice comparison chart for American Wire Gauge to IEC 60228 in this file:
- Comparison of SWG (red), AWG (blue) and IEC 60228 (black) wire gauge sizes from 0.03 to 200 mm² to scale on a 1 mm grid – in the SVG file, hover over a size to highlight it
You surely can, but the connectors (not the controller) are the weak spot on this machine (especially combined with the lack of strain relief on this machine). Many people will prefer longer cables to avoid connectors in the middle, because the more connectors of this kind you have, the more possible causes for bad connections.