and how good are these compared to either straight sword, or arabian style sabers and falchions?
Huge disclaimer: I'm not trying to be uber pedantic, yes I know almost every term I'm using isn't exact, historians of civfantics, plz no persecute for incorrectly comparing pollaxes to halberds etc.
While a bunch of stuff goes into sword design, almost everyone in the europe-ME-asia part of the world had swords designed for a particular purpose. This is why there are so many types of swords even at any one time: they do different things.
Curved blades in particular (a la scimitar, or a katana) are for specific type of cutting. If you want to just hack something in half meat-cleaver-style, then curve isn't that important. But rarely are
swords cutting people in half: Slicing someone open is also very deadly. So here enters the curved blade: draw cuts, as they are called, are basically where you
draw the blade along your target. Kind of like how you might slice a piece of fruit in one clean motion. When might this be useful? Well, against anyone unarmored. And if you swing your blade from horseback, it's going to have some natural drawing as you cut through hapless foot soldiers. (Think of how 19th century cavalry sabers are depicted.) The more armored your target, the more useless cutting is and the better thrusting becomes. Thrusting wants a straight, pointy sword. This is why we associate the "middle ages broadsword" as being a straight blade: lots of steel armor in europe.
Then you have sickle swords. The outward protrusion of the blade makes it act very similar to an axe: it's good for chopping. Contrary to what you may have been lead to believe, axes are extremely brutal and effective weapons, the downside is only the axe head is deadly. So if you make a sickle sword, you still get the full blade, but also something like an axe head for chopping. The idea that a
ngulu would be used for execution is logical, then: chopping is key for a clean decapitation. It isn't even hard to see why so many cultures (khopesh, ngulu, the kukri) came up with this idea: if it works for cutting trees, it probably works on people.
Back when people in the middle east used sickle swords, the types of of shields and armor they wore made hacking and chopping very efficient way of killing people. Egyptians are often depicted with axes as a main weapon. in fact, some people think the khopesh evolved from the axe:
It's not to say any sword is much better or worse than another; even ancient people aren't stupid, and if they were, they didn't stay stupid for long. They all evolved to defeat whatever they are facing. Many swords can do several things well, but just to boil down what the various blade shapes are specifically good at:
Straight = thrusting
Curved back = cutting
Curved forward = chopping.
Side note: It's not like the khopesh or ngulu's "axe chop" concept were weird one-offs outclassed by later swords. As the middles ages progressed and steel plate armor got really good, knights on foot wouldn't fight each other with swords - they used pollaxes. (Think halberds.) Even though an axe won't cut through steel plate, you can definitely still batter the guy inside and break his bones. (And then stab him in the eye with your dagger.)