I finally purchased a quality straight razor, a W&B, from someone in shave-ready condition (the razor I mean). I already have soaps, brushes etc., but I'm wondering about stropping and also what else I should get, now and/or eventually.

I actually have an Illinois strop from an ill-fated attempt to get a straight razor in the 1990s:

Name: IMG_20140710_140617.jpg Views: 347 Size: 34.1 KB
It's an Illinois 835, but it looks a bit different from the 835s I see for sale online.

I also have a piece of wood with leather fixed to to both sides (the leather's about 3" wide) that I have for sharpening knives:
Name: IMG_20140710_140534.jpg Views: 346 Size: 51.3 KB

I have the Illinois strop currently hanging from a screw in my bathdoor door (set about head-height). That's probably not ideal for stropping, no? What is the best way/place to attach this sort of strop?

How would using my leather-fixed-to-wood strop compare to the hanging one? is it worth reserving the former for compounds? or is it better just to get balsa wood for those purposes?

FInally, other than stropping (and occasionally using chrome oxide?) should I be bothering about doing other things to maintain the edge? I think for the first time it needs rehoning, I'll probably send it out. (Though eventually I'd like to learn how to hone). Is it worth picking up a very fine waterstone or film for mild rehoning?