What gives samurai sword its strength?

Answer 1

It is made of steel that is folded 1000 times

This is incorrect. The Japanese never folded a blade more than 15-16 times, 16 folds creates 2 to the 16th power layers of steel in the blade. This is more than 65,000 layers. Folding it any more than this creates a blade with an homogeneous effect, which is what you started with in the first place, before it was folded at all. This is why they did not go any further than 16 folds because it is counter productive.

The samurai swords strength and weaknesses depends on the type of blade with which it is made from. This is the major factor which plays an important role in identifying the quality of a real samurai sword.