I've TASed the beginning (up to about 5 minutes in) of Paper Mario: TTYD on three different Dolphin versions (r7323, r7719 x64, and 3.0-191) and I still haven't gotten any desyncs. You probably don't want to start a game on one version and try to continue it on another version; it will either desync or erase your changes and revert to the old movie file. If you move, say 100 or more revisions ahead, you will probably have to restart.
There was recently a "savestate fix" that apparently fixed all issues with savestates, and hopefully problems like the input backshifting MUGG described. I'm not sure if it helped prevent desyncs or not (as the only game I've tested never desyncs). The fix is implemented in every Dolphin revision since 3.0-183.
The TAS input window now has checkboxes for the GameCube controller buttons. However, there aren't any graphics or click-and-drag additions for the control sticks, and the game won't accept button presses from the keyboard or joystick when the window is open, only the TAS Input window. Also, the input window currently copies its input for all controllers that are plugged in (no 2-player runs can be done yet with the input window). But if you can look past all that (especially if you NEED it to optimize your game, like I do), and you just want a basic input window, it works great.
Also, according to
this page, SMG2 is almost perfect, if not perfectly emulated. I don't know if it's sync-stable, but you can try it out and see.
