Elcb keeps tripping even with all MCBs turned off

If you are correct that nothing on that circuit is working (including an outlet closer to the circuit breaker panel) then something would appear to be wrong at the panel.

I wonder if perhaps you got the wrong breaker and the actual breaker for this circuit is sitting in the tripped position? Sometimes it can be useful to flip each breaker on and off one at a time (of course that can play havoc with whatever else is on those circuits, such as computers wot don't have a UPS).

If it was only that outlet and ones further down I would suspect that the incoming wires are not correctly connected or the outlets have been separated by breaking off the tabs between the two receptacles.

I agree with ppa about the wire running down the stud. Try to determine which side of the stud it is on by slowly moving your detector away until the light goes out. Probe for the stud with a 4d finish nail until you locate the exact ouline of it. Then follow bats advice and carefully drill the stud with a pilot bit for your lag bolt. You may want to be just a hair off center away from the cable but still in the meat of the stud. The pilot bit should be half the diameter of the screw or less. This should guide the screw and keep it from wandering off.

If you take your time and are careful you should be able to avoid the electrical cable in the wall and still get the holding power of the stud. I would not trust #10 screws or drywall anchors for this job.

The best way would be to remove a part of the drywall, attach horizontal blocking between the studs and patch the wall back up. Then you have a really solid installation. But then, this will cost more time and money.

Good luck!