Would it be safe to simply carve a 1" groove out of the drywall, tuck the wire in to the drywall and use a staple to keep it in place, then use drywall tape and mud to patch it up?

No!

And a picture or two will get you the best suggestions.

The proper way to remedy this half assed installation is to pull the wire back into the attic, or whatever space in the ceiling the circuit is coming from, drill out the top plate above the stud bay the circuit needs to be in, and fish it down properly to a rated cut in box. If you don't understand what I'm explaining, then my advice is to call a reputable electrician to correct it.

Jason-

What you describe as a remedy is a clear code violation, and dangerous to boot. You need the end of the cable to terminate in a proper electrical box, and you can't have cable buried in the wall close to the surface where someone could easily put a nail or screw through the hidden wiring. Get it fixed correctly as described above.

I haven't been in her attic, but the best I can tell it looks like the contractor spliced a wire somewhere, threw it across the ceiling, cut a hole in the ceiling with a knife, and dropped the wire through.

And that's about par for the course on the work they did. I'm meeting with the Sheriff's office on Wednesday to determine whether this is a civil or criminal offense.

They took pretty much every penny she had, so at this point it's up to the family to get everything fixed. I would LOVE to hire a pro to fix things, but it's just not an option anymore :-( So I apologize in advance if I have to keep asking for advice!

you can't have cable buried in the wall close to the surface where someone could easily put a nail or screw through the hidden wiring

@kudzu9, I'm not entirely sure how to avoid that.

This is a cinder block house built in the 1950s (I don't think it was originally intended to be a house), and the walls just have 1x4s mounted to the cinder blocks and then drywall mounted to those 1x4s; so there's a 1" gap between the cinder block and drywall, filled with insulation board.

I can't imagine a way that ANY of the wiring is protected. At least, not on the outside walls.

If there is a wire splice that's not enclosed in an accessible electrical box, that's a code violation, and potentially dangerous, too. Based on your description, I'd be willing to bet this scam artist was not a licensed electrician and didn't know or didn't care what he was doing. As far as having safe wiring, the way this is typically done in the situation you describe is to run all the wires through metal conduit or use a product called Wiremold which is a protective channel that the wires run through that is surface mounted on walls and ceilings. I suspect from what you say that the house is highly non-compliant with code and may never have had an electrical inspection. I'm sorry you're having to deal with this, but I recommend you at least hire a qualified electrician for an hour to assess the wiring situation in the whole place, and get an idea of how dangerous the entire situation is.