Testing swords on straw was a common method of demonstrating the sharpeness of a weapon, as straw is very light, if the blade were dull it would bend the straw rather than cut it.

The tempered edge of a weapon was very hard but good swordsmiths would fashion a sword in layers with a core of softer steel/iron at the center of the weapon. The softer steel absorbs some of the impact on a cut preventing the weapon from snapping in two, which it would do if only made of a hard steel.

Swords were expensive weapons, and ones made by noteable swordsmiths would be expensive indeed. The edges were polished with stones of varying hardness and roughness until the edge was polished and sharp. There are many beautiful examples of the swordsmiths work on exhibition in the Armouries Musuem in Leeds, Tower of London and in many of the UK's castles.

A bent sword could be straightend by a good smith. Cheaper edged weapons were often just sharpened on a grindstone with no particular regards to finish.

In Japan a new sword was often tested on a dummy made of wetted straw tied around a bamboo pole, a makiwara it was called. Varius cuts would be made as the makiwara had the consistancy of a human body. This art of cutting a makiwara was called tameshigiri and tested the sword owners skills in both drawing and cutting cleanly.

I would think that much the same procedures would be adopted by other swordsman in other country in order to both imporve the strength and skill in order to stay alive in a fight.