Which Fabrics are suitable for origami tessellation?
Cottons which take folds well and hold the crease when you rub or iron it in.
I would look for a woven fabric and not for a certain weight but I would experiment with several which I have around in the house.

I know that sail makers used to use a sail rubber when having made a fold in the sail before sewing.
Sail cotton (also called cotton duck) is about as heavy as you can get it and still be able to work it by hand.

Whether you need to stiffen it depend on what you want to do when the initial folds are in and on the quality and weight of the fabric.

Some other fabrics will work, if you use an iron or steam in the folds. Traditionally woolen fabrics were used to make pleated skirts where the folds worked pretty much like the patterns you plan to make. Steaming those fabrics was a specialist job, but simple patterns should be within reach for people at home.

The best is to experiment.
And please come back to tell use what worked for you.