Shape, not much. The area enclosed by the winding is what's important. Square or oval or other shapes, are just a matter of convenience, maximizing useful area in a given form factor.
Coupling distance is on the order of the diameter, give or take how low of a k (coupling factor) you can tolerate.
Litz is used because the Q factor needs to be relatively high for reasonable efficiency (e.g., ballpark Q = 100 for k = 0.1 and efficiency <80%). At power conversion frequencies (i.e., 100s kHz), solid magnet wire will get Q in the 10-100 range; Litz can do about 10x better.
If you have lots of fine magnet wire, you can make your own Litz by bundling up a bunch. Just a bunch of (7 or more) strands will help, or you can make rope by twisting multiple bunches together (which is what proper Litz does).
Planar magnetics (PCBs) can be okay but it depends on how much effort you put into them. A structure resembling Litz can even be made on a multilayer board, but it of course takes up a lot of space, and is a bit of a pain to draw. The aspect ratio tends to be unfavorable (flat traces are "all corners", versus round wires that allow magnetic field to flow around them more easily), so you can't get the same performance as a wire coil of comparable dimensions (i.e., same copper cross-sectional area, different profile; same winding pitch, size, shape and turns).
Tim