Simon, yes all ESCs cut out when the battery nears the allowable limit. They often even have two styles of cut-out -- hard and soft. Soft means the engine slows down (to let the pilot know before cutoff that the motor will be cut. The hard cutoff just shuts the motor down.
When you set the programming for the number of lipos you are also setting the cutoff voltage.
Yes you can program an ESC with just tones and the throttle. It can get confusing at first, but once you "get it" it's do-able. I've never bought a programming card, and do all ESC programming "on the beep".
One more reference that might be helpful for programming this ESC, and seems to confirm is the link you gave me to Hobbyking for the "similar" esc.
If you go to that page, toward the bottom you'll see a set of tabs. Click on the FILES tab and you'll see support files for that ESC. The top two files (numbered) are in English and are instructions for programming that ESC. Download those and compare to the link I gave earlier.
Now as for actual settings -- Suggestions:
Set the number of cells in series you will have. Set lipo. Set no brake. You want probably soft start (usually used on helicopters -- a gradual ramp up n speed). Try auto timing. If that doesn't work (loud squeal when you crank the throttle) try soft or lower timing.
But please figure out why you had that short, if you haven't already. Was it really moving a bare lead or pin to contact something accidentally, or was it wiring hookup error. If the former, then things should be better now that wiring is soldered and insulated, etc. But if wiring was wrong, it will happen again (unless it was changed).
It looks pretty definite that your ESC has no on-board BEC. It therefore does not need the center (red) lead going to the servo tester, and absolutely requires a separate battery to drive the servo tester (assuming it still works).
It would be a good idea to test that servo tester somehow separately before trying to drive the ESC and motor from it. Or buy a new servo tester to be sure. They are the least expensive component.