More progress! I finished sanding and staining this weekend. I'm using "American Walnut" water based stain by Minwax, and the water based "Satin" polyurethane. I started with the bottom seat piece first to hide any mistakes. At first I was painting the stain on and then wiping it off immediately with a rag in patches. This caused overlap marks and was inefficient - I figured out that if I quickly painted the entire piece and then wiped from one side to the other in straight smooth motions, the stain came out dark and even. (duh!)
The polyurethane went much better - except for the edges, if I wasn't careful to pick up the excess with the brush, it would pool. I did 3 coats, and lightly sanded with 220 grit paper in between. The result is just as I'd hoped - dark, rich walnut, it's really going to look amazing with the grey/green vinyl.
Next step is to stitch rip the old vinyl to get the welting cord - since the cord is the correct length, why bother cutting it new? I also lucked out at Jo-Ann's, they were having a 50% off sale on foam - I bought three 2" thick pieces to add to the old foam. The existing foam is in fairly good shape, but could use a little extra cushion.