Pretty similar to your method but I don't rinse before sanitizing.
New (or neglected) bottles go into an oxyclean bath (labels, dirt), brush them out (if required), rinse with real hot water and stick them on the tree. Once dry I store them in the cardboard cases they came in. I put a double layer of paper towel in the bottom to soak up any final drips and stick the bottles in upside down. I've lost a couple of bottles doing this through clumsiness but only had one that I couldn't get clean. When I started home brewing I got a bunch of empty bottles from some family members. Some of them looked pretty sketchy (bottles not family members) but oxyclean and elbow grease got them all sparkling, except one. I don't know what it was down there (tar?) but it defeated me and the bottle found a new home in the recycling bin.
For my bottles that already have homebrew in them I just rinse with hot water and drain on the tree before putting them back in their boxes. The key is to do this as soon as possible, right after the bottle is emptied is best.
On bottling day I just sanitize the inside of the bottle and the top of the neck using a star-san in a bottle washer and then drain on the tree. I don't rinse before sanitizing because the bottles are clean and since they are stored upside down nothing can fall into them.
