The curling up has a simple reason: the burning was not uniform. Paper contains water (really!), and heating it makes it lose the water.
Now, paper is a bad conductor of heat. When you burn it, inevitably the flame will be mainly on one side(the side that's pointing up). The flame will transfer heat to nearby regions of paper, and evaporate some (or possibly all) of the water. The loss of water leads to a contraction of the cellulose fibers in the paper (I'm not too sure of this, comments appreciated). But, this only happens on one side of the paper. Even though paper is razor-thin, the fact that it still has two "faces" is important here. Now, due to the contraction, the paper curls in that direction (similar to how a bimetallic strip works). That's it.
The reason for retaining texture, is something I'm not too sure about. The burning reaction throws up a lot of soot, but a lot of the products stay right where they are. This results in the charred/burnt paper occupying the previous position of the cellulose. While charred paper isn't a polymer and thus isn't too strong, it can "mimic" the initial structure just by occupying the position of the cellulose. But, unlike cellulose, if we try bending it, it will crumble since there are no long polymer "chains"/intermolecular forces holding it together. It's like the difference between a house of cards and a similar "house", made of a single piece of rubber. The structure is the same, but the former case is only staying together because it had the "luck" to exist in a stable configuration. Disturbing it will destroy it.