GE fridge started buzzing, is it safe to unscrew the panel to clean coils?

Unplugging the unit is the first step. You're totally fine to take that back panel off; it's just a dust shield and access point. Since you mentioned a thick layer of pet hair, getting a vacuum in there is actualy neccessary to keep the compressor from overheating. If those coils can't shed heat, the compressor runs longer and hotter, wich eventually kills the start relay or the motor itself. The buzzing you're hearing is usually just metal-on-metal vibration rather than a mechanical failure. When the compressor kicks on, it vibrates at a specific frequency. If that metal panel or a nearby refrigerant line is even slightly loose, it'll rattle like crazy. That blue tape you saw was probably the previous owner's low-effort attempt to stop a rattle. Be extremely careful with the edges of that metal panel once you get the hex screws out. Those things are often **razor sharp**. I've sliced my knuckles more than once trying to wiggle them back into place. When you go to put it back on, try slightly bowing the metal inward. This creates tension when you tighten the screws, which stops the panel from flapping against the frame and solves about 90% of those buzzing sounds.

Get a long, thin crevice tool for your vacuum or even a dedicated coil brush. Don't just suck out the loose stuff; you realy need to get the hair off the fins. Just be gentle with the copper lines. If you see a small fan near the compressor, make sure that's clear of debris too. If that fan is gunked up, it can't move air over the coils, and that can cause its own set of groaning or clicking noises.

That bit about the sharp edges is a lifesaver, I defintely woud've sliced my knuckles. Since I’m already planning to get in there, is it normal to find a plastic tray filled with gunk and water near the bottom? Mine looks pretty nasty and I can't tell if that's a leak or just where the defrost water goes. One more thing—is there any danger of static from the vacuum nozzle messing with the fridge's computer? I’m worried about hitting a wire and shorting somethig out while I'm trying to get the dust bunnies.

That plastic tray is just the evaporator pan. Defrost water drips down there so the heat from the compressor can evaporate it. If it’s slimy, give it a quick soak in some soapy water so it doesn't start smelling up your kitchen. Don't worry about static from the vacuum nozzle; it's a non-issue around the compressor and coils. While you have the panel off, give the condenser fan a spin with your finger. It shold move freely. If it resists or makes a grinding sound, the motor bearings are probably shot, wich is a common source of that buzzing. Sometimes that blue tape is just used to keep wires from rattling against the frame, so check for any loose harnesses.

Since you're alredy in there, check the rubber grommets under the compressor. Over time, those little rubber feet get hard or brittle, and they stop absorbing the vibration. If they’re cracked or flattened, the compressor just rattles against the base tray no matter how clean the coils are. It’s a cheap fix if that turns out to be the source of your noise once everthing is scrubbed down. Just be careful not to lift the compressor too high while checking them, or you might kink a refrigerant line.