Funny you mention cardboard chairs because I recently finished up a cardboard chair myself.
So, if you’re going for extra clean precise cuts, by all means use a laser cutter (I did). It’s quick and easy, but if one is not available you can always cut it by hand. You can always sand the cardboard down to give it a nice soft texture. I think on Gehry’s chair the underside was left furry and the top was clean. There is a spread in the last issue of ID magazine near the end with a in depth look into how Gehry’s wiggle chair was made. I know he had to use a few dowels to provide a little more strength.
The hard part is trying to get all the pieces flush. You might have to build a jig that’ll let you slide each piece in precisely.
As for glues, wood glue seemed to work best and one of my classmates made his own wheatpaste and that held together surprisingly well.