Is it possible to access tortoise from the system tray? Can I create a system tray icon for a repository?
Depending on your version of Windows, you may be able to put folder icons in a "quick launch" section near the system notification area (the "tray"). Then, you can right click on these folder icons to get to TortoiseSVN just as if you right clicked on them anywhere else.
I've done this with Windows XP, but I haven't used Vista or later so I don't know what the procedure is to put folders somewhere on the task bar.
It's not possible to access TortoiseSVN from the system tray, because that's not the way it's intended to work.
You are expected to work in Explorer; any right-click anywhere in the window will bring up TortoiseSVN.
Personally, for me, it's a very intuitive way to work, as I mostly begin working by navigating to my woring folder, and then launch my editor from there, say by opening a .sln file, rather than starting Visual Studio from the start menu and opening my project from the Recent menu. It gives me a feel for the layout of the project and the building blocks of my work. I don't really know how to explain it any better than that.
But in such a workflow, TortoiseSVN integrates perfectly as it is.
You can have something like you described using SVN Monitor. It's free and it allows you to define various repositories. You can also commit via the system tray if I am correct.