To be sure of a sound attachment, you need to attach into the concrete. Anything else will almost surely fail. Plaster does not have the structural strength except where you can spread the load over a large area from behind the plaster, and even then it is iffy.
As suggested by others, to attach to the concrete, you can use tapcons, concrete expansion bolts, or expansion anchors that you put screws into. Any of these will make very small holes in the concrete and not compromise its strength. These holes could easily be filled when you leave.
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You need to know what you are drilling into if you go that route.
A very different approach would be to create what is basically a free standing rack and then fixing it lightly to the wall.
The existing rack could be screwed or bolted to the top of the face of 2 upright hardwood boards, maybe 1x3s, as tall as you need the rack to be. The bottom of those uprights would be attached to a foot extending out from the wall about 12 to 15 inches. A brace would be attached on a 45 degree angle from the far end of the foot to a spot on the upright about 12 inches up.
The braced feet will counteract the downward pull of the coats. You could even attach a flat board to the tops of the feet to make a shelf for shoes, etc.
This whole unit could then be screwed into the wall through the uprights into plastic anchors in the drywall. This is just meant to keep it from shifting, not supporting the weight of the coats.
This sounds like a rather dangerous situation. The drywall installers no doubt did not install metal straps on the studs wherever an electrical wire crossed through the stud. Most likely a drywall screw or nail has gone into the stud and entered a wire that crossed the stud in the same place.
The fix for this may require a professional electrician unless you are pretty familiar with electrical wiring work. It will be necessary to cut the power to the circuits and open all the applicable electrical boxes and disconnect the various lights, switches and power feed wires. Then shorts checks can be made using an appropriate ohm meter for all the wires in the area between the boxes.
Once the short is located it will likely be necessary to remove the drywall in that area to replace the wire.
Do not leave this problem go. It sounds like you have a medium resistance short in some wire/stud crossing. I say medium because you did not mention that it is tripping a breaker. The resistive short can produce a lot of heat and could definitely lead to a fire in your wall or ceiling.
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If your hollow walls do not have wooden studs, the studs are most likely steel (at least in the US).
You are right to be concerned about the extensive wiring near an electrical panel. Numerous wires may be exiting in virtually any direction. You could check out where wires exit by turning off the main power switch and then removing the panel face plate. You will be able to see the directions most of the wires follow. Caution! There is still live power coming into the main breaker, even when that breaker is off.
If the wiring is up to code, cables are supposed to be recessed from the front edge of the metal stud by at least 1.25 inches (NEC 303.4(D)). This leaves you some room to drill and install anchors in the drywall, although I would avoid drilling very close to the panel (as you suggest).
For extra safety, you may wish to turn off the power when you are drilling on that wall, take care not to allow the drill to penetrate much beyond the drywall and, if you seem to hit anything other than drywall, check to make sure it is not a cable before progressing.
You can mount things on steel studs, but you need special screws, not wood screws. Also, if mounting on steel studs, be sure the screws are not so long that they penetrate the steel more than about 1/2 inch at most (longer provide no additional holding power in steel studs).
While steel toggles and expanding anchors should not hit wiring if it has been properly spaced, consider the plastic expanding type if you want additional safety.
I would avoid the tubular type plastic anchors except for the lightest of loads. And do not use the steel hammer-in type anchors!
Note: If you think you might have hit a cable, be sure the power is off and call an electrician to check it out before proceeding!