Partition corruption, still can see files in single user mode, would like to retrieve files
If you have some understanding of partition layouts, you could try using my disk editor iBored (http://apps.tempel.org/iBored/) to check the drive and its partition, search for the partitin start, then maybe copy just the partition data to a disk image file or to another disk's partition. All this can be done in iBored.

You could also try using iBored to erase the initial partition blocks (and save those blocks to a file first, so you could later restore them), to see if Data Rescue is then able to read the disk without hanging.

Also, you describe that in Single User mode, you tried to mount the (damaged) root disk as writable - that's not possible, clearly (since the volume is damaged). But have you tried to mount an external disk in writable mode? If that works, you could easily copy your files over then.