For the razor storage a snap close case with some silica gel. The pelican micro cases might be worth looking at. Specifically the 1060, 1050, 1040 models. Just make sure you measure the razor and pick the box that fits.
1040 Protector - Micro Case Seriesā„¢ | Camera Cases | Pelican Products, Inc.
Silica gel can be had off ebay. There is one type that changes colour when saturated it's probably the best one for you. It can be dried and re-used. It'll keep the inside of the case dry and the razor rust free.
Rock wise. Coticule if you can get a decent one in the right size. 150mm x 40mm would probably be the smallest I would personally want but you can get smaller. The 1040 case is 165mm internally and should fit a 150x40 stone. A standard razor card box is also about 165mm long so it should all fit really well together with perhaps space for another razor or some extra bits like a small pasted balsa strop.
If you are adamant on a synthetic stone they do cut. You can make a norton 4/8 into a 165mm stone you'd have to saw off about 40mm as 8 inch is 203mm. You could also do the same with the Naniwa 3/8 and have the advantage of a splash and go stone. Unfortunately both the naniwa and norton absorb some water in use and neither can be wiped dry. So storing the razors with a freshly used stone might saturate the silica very quickly and make it useless till dried in an oven. (Coticule is generally wipe dry as are a lot of finishing naturals).
Barbers hones are something i've never used. Modine has a fantastic article on another forum that goes through a vast collection of them. And it might be worth looking through that and searching for a specific brand. I'm not sure how absorbent they are, And I'm not sure about cutting one up as they have different production methods and binders depending on the make and model.
Barber Hone Reviews