Why can a piece of paper be folded along a straight line but not along a curved line?

No, paper can't be stretched because its structure contains no slippage mechanisms which would allow it. To some extent, you can provide slippage by wetting the paper into mush, which unlocks the entanglements between the paper fibers and allows the sheet to deform.

Apparently, while you can't fold paper along a curve in 2 dimensions, you can do it in 3 dimensions.

This article from the journal Physics describes how a flat ring of paper, folded along its circular center, deforms into a saddle as the material buckles to absorb the stresses.

(Longer discussion at Physicsworld courtesy of an answer to a related question.)

The short path between 2 points on the paper before folding is a straight line, all points of which belong to the paper.

Now suppose that we managed to completely fold it along a curve between these 2 points. All paths between them on the paper are now curves, with bigger lengths compared to the straight line, (which passes out of the paper).

So, it is not possible to fold it along curves keeping the same distance between all points. And it requires streching to change distances.

Paper can be more-or-less folded along a curve. You can do it with your fingers.

It could be argued that this is really just a lot of little short straight lines.

But it could be argued that any curve is just a lot of little short straight lines.