In short, the problem with your categories not transferring should not be an issue since Steam actually syncs all of your category information to the cloud. It is a matter of patience in giving the alternate installs of Steam on your secondary devices a few restarts until they show up. By resetting your categories right away when you see they aren't there you are basically giving the sync another version of categories to sync up and it is simply over-riding everything you did before with an exact copy. So instead of re-establishing the categories, just keep restarting the client until they show up from the primary device.
Unfortunately when I say "SHOULD NOT BE AN ISSUE" I mean to put emphasis on should. It doesn't always work to perfection. Just a fact of life at this point.
As for your secondary question of being able to order your games in a more organized manner you really have a limited number of obvious options for this:
- View Options Selection: Detail View, List View, Grid View (I prefer List View)
- General Sorting: Clicking on the Column you want to sort by.
- Favorites: Selecting which games you are playing at any given time to make them favorites.
- Categories: Which you are already familiar with, naturally.
I personally find categories to be more trouble than they are worth and can make my game list more intimidating than it actually is. With 210 games currently on my Steam library, it is much easier for me to simply manage my list by games installed vs. not installed and then to further that of the 15 or so games I may have installed I keep about 5 set as a Favorite. I never go over a certain amount of games installed and I usually decide which of the games I want to uninstall to make room by sorting them by which games are cloud connected. That means if I uninstall it, the save files remain saved for me and will return when I re-install the game and come back to it.
I know it's easy to say, "Categories organize my list and organizing my list makes it easier for me to navigate" but maybe you should delete all of your categories and give a category-less list a shot for a month and see how it goes.