Cute fabric!!!
There's not a whole lot of difference between cotton and polyester. There's no point in pre-washing polyester (not that you'd want to, with the squares already cut) as it doesn't fade or shrink.
It is more slippery to work with than cotton, so can be a little harder to match seams. I would probably spray starch before sewing, in hopes of adding more stability to the fabric.
Polyester frays more easily than cotton, so you might want to consider using a 3/8" seam or a full 1/4" seam rather than the more common scant 1/4" seam. If you are just going to piece the squares together, that won't create a problem.
You can use any kind of backing you want; it doesn't have to be polyester. You would probably want to pre-wash the backing if it is cotton, though, to make sure it won't shrink more than the top. Just be aware that cotton fades over time; polyester does not.
You can quilt or tie; your choice.