Baldur, I don't believe you are experiencing any "one" problem, rather, you are are experiencing many problems at different times.
If I'm reading correctly, you listed each incident as you experienced it separately of each other. Your description of the problem is excellent and may help you find the solution to your "problems" faster.
First off, you said the problem usually happens when all "sound" effects are muted, and I noticed in your listed system specs that you are using a GTX 560 Ti which supports streaming audio with an HDMI cable.
Are you playing the game connected to an HDTV or converter that has it's own speakers built in? If so, it could be that your "video card" drivers are partially corrupted, as the sound drivers for the Nvidia 500 series cards are built into the video card drivers or installed "with" them.
It could also be that the portion of your card that controls the audio is damaged.
(please disregard this information if you are not using an HDMI cable or adapter).
Second, you said the next time this problem happened the events were almost exactly the same, except that your character was moving in circles and you could not get a response from your "Xbox 360 controller." You had to resort to using your keyboard, which also was unable to move your character. You mentioned that you "panicked" and attempted to "restart" your computer by "pressing" the power button.
Now before I get to the power button bit, I'm going to bet that your Xbox 360 controller is wireless, and that you panicked "before" you lost response from you controller. I'll bet you either made a sharp motion, or dropped your controller when a familiar, and disturbing problem popped up "again."
Whatever you did caused your battery pack on your controller to momentarily become disconnected from the controller (Note: disconnected does not necessarily mean that the pack fell out, but that it received trauma that caused it to stop providing power to the controller). Your controller shut off and your character was stuck in a command line that had been transmitted to the game before your controller went down, thus the looping motion. Re-powering the controller, at this point, will not give you control of your character, because FFXIV does not re-associate wireless controllers with the game.
You have to log out and log back in, or have a special driver installed and configured for your Xbox 360 controller.
You mentioned you tried to Alt+Tab out of the application?
That was the correct reaction to actually "fix" your keyboard not working with the game. I've actually experienced this problem with my own computer system.
It will NOT work, however, if you do not have something (a folder, another application on the system bar) running behind the game.
Sometimes FFXIV will not recognize your keyboard... I have no idea why this happens, probably something to do with the game pad again. Alt+Tab will fix this if you tab out to a folder or another application running in the back and then tab back to FFXIV (You may have to do this a couple times to get it to work). You should regain full control of your game with the keyboard if this works.
If you're not running something like ffxivwindower.exe or FFXIVEvo that will window the game when you play in full screen, this will obviously crash the game and not fix anything (Once again, will not work unless you have something running in the background) but it will succeed in getting you out of the game.
About the power button: I'm going to just assume you're like most of us who play this game and you had to build a high powered "new" PC, and before this, you've never had to restart the computer using the power button. Many new PCs or systems require that you set a parameter for resetting your system. If you don't set this in the Bios or in Windows, it automatically assumes that a simple "press" of the power button means nothing. However, holding the button will always mean to turn off the PC.
Finally, your "R0" issue.
Believe it or not, the game will not "always" give you the red R0 when you are experiencing it. Everything will freeze on your screen, other characters, monsters, animations, but you will still be able to move your character. This is a temporary freeze and anything you do between the time it freezes and the time it comes back, will fast forward in a rush when your connection resumes, which is why your logout window popped up after a while.
The reason your character became "stiff" is a known graphical issue, and not at all related to your graphics card or system, but is related to bad coding by the developers.
It will usually fix itself, but I've experienced it before where it would not go away until I restarted the game.
Sorry for the extreme wall of text, but I felt your accurate description of the problem deserved an in-depth explanation. I hope the information here helps you solve your problems, if not, sorry I couldn't be of more help.
One last question, what video card drivers are you using?