There are a bunch of things going on.
--Edge geometry depends on what you're chopping. One of the forumites very kindly reprofiled one of my khukuri knives after I chewed the edge up chopping wood with it. As originally made, the edge was too thing for the amount of force I was putting on it. In general, wider edges work better for harder targets, while narrower edges work well for soft targets.
--Steel depends in part on the source (side note: I thought we had a link about steel types, but I don't see it any more. Anyone have it?). In general, places like Scandinavia and Japan depended effectively on bog iron which they had to forge themselves. This gave them a fairly complex mess of material, and that material included a fair amount of blade-weakening silicon in the metal.
To build a sword out of such a mess, you chop the mess into bits with different properties, from high carbon edge material to low carbon back material, weld these bits together into what will become the blade, then fold it repeatedly, not just to make for a stronger complex joint rather than a single weld, but also to control the size of the silicon crystals. Too big a silicon crystal would seriously weaken the blade.
Also, as with the keris, patterned steel is highly decorative, so this process can be deliberately controlled to produce patterns, as with keris blades.
Obviously, if you have a uniform steel source, especially if it's low silicon steel, you don't need to work the material so much.
As for edge hardening, I suspect that depends again on what you want to do with the blade. The argument here is that harder edges are harder to resharpen, and depending on what you plan to do with your blade, building it to be easily resharpened (as with a Swedish mora knife) might be more desirable than have it get dull slowly, then be a real pain in the ass to resharpen, absent a diamond block (as with some modern steels). Also, it's not clear how much different smiths knew about differential hardening. While I'm not a smith, differential hardening doesn't have to be as complicated as the Japanese make it to be. Khukuri makers do it by hand, with a kettle of cold water, and make nice hard edges on their khukuris.
Curvature is a whole other layer of complexity, and I'll let someone else take a swat at that.
Best,
F
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