This is mostly a 'me too' post. Tharkis, I don't have a definite solution to your problem but don't lose hope - I can (inconsistently) mount my S3 on Mint 13. If I could figure out how to consistently mount it, or where the bugginess lies, I'd be much happier! I've googled around the subject but never got to the bottom of it... but I'll share my experience and it might give you a few things to try.
Firstly, I'm using a rooted i9300 S3 (i.e. the international version), running Cyanogenmod 9 rather than stock Samsung firmware. CM9 is based on Ice Cream Sandwich. This may or may not be relevant.
I'm assuming you have already installed mtp-tools and mtpfs - supposed to give us mtp support in linux. If you don't have these, firing up the Software Manager and installing them would be a great start. That said, in my experience most mtp devices are a littly flaky in linux, not just the S3.
I also had your experience with Banshee locking up (with the S3 in MTP mode), and the same with Shotwell (with the S3 in camera mode). Like you, I eventually disabled autorun. I'd still get a dialog box with the error "Unable to mount GT-I9300 / Error initializing camera: -60: Could not lock the device". After dismissing the dialogs, I usually find two GT-i9300 devices mounted (the phones internal storage, and it's external SD card) but similar to you, can't see past the root directories.... or actually copy files, or do anything useful.
If I switch the S3 connectivity to camera mode, and then back to mtp mode (via the Android notification drawer which pops up a "Connected as a media/camera device - touch for other USB options" message when you usb connect the phone) - it will usually mount properly in Nautilus after a variable (maybe 30-60 seconds) length of time. I can see and interact with the whole filesystem of the external SD card, and the parts of the internal storage which mtp lets you access (i.e. not the parts that you need root access on the phone to see, but this is expected).
This doesn't work 100% of the time - I can't figure out why. Sometimes I connect/disconnect and switch mtp/camera several times before it mounts properly. Some people claim turning on Android debugging (system settings > developer options > Android debugging) helps, but it doesn't make any difference to me. Interestingly, attempting to connect the phone as a camera (ptp mode) never works. Also, I can USB-tether the phone and use it for interweb access, but can't mount the filesystem at the same time.
Lastly, if none of the above helps, you could try the gmtp tool from the repositories - this has worked with every MTP device I've tried. It's a clunky interface and pretty slow to up/download, but it does at least give you basic access to the files on your phone.