Also, a chicken hung up in electric poultry netting that is not touching the ground is not getting shocked. Same as a bird sitting on a high voltage power line does not get shocked. Both you and it would both get shocked if you tried to untangle it when the fence charger was on and you were standing on the ground to close the circuit.
A hot, charged fence is only a potential shock. The current flows from the fence to the ground when a connection is made through a material that will conduct a charge. When something touches both at the same time, it works just like a light switch that allows the current to flow. Grass, etc, will make that connection. So will you or a chicken or a predator. But you, in rubber knee boots, may not feel it. Lady I'm married to walked into a super hot fence once, but lucky for me, was wearing thick, puffy flip flops at the time, so was insulated. Otherwise, I would be dead right now. And not from the shock, but from blunt force trauma from the beating I would have gotten.
On your landscape cloth, unless it will conduct electricity, an animal standing on it will be insulated from the shock, so they may not feel it. Depending on how wide the landscape cloth is, a coyote, dog or fox may have their front legs on the cloth, but perhaps back legs standing on the ground. If so, they might get it.
Paper feed sacks or even newspaper might work to keep the weeds down, yet still serve up a jolt especially once they have gotten wet.
On your ground rod, most instructions for how to install those are vast overkill. At most, a foot of so of rebar, or even a rusty steel T post, driven in the ground a foot or so will do it.
With poultry netting, the posts they hang the stuff on do not conduct electricity, so it many cases, you can brace the corners to keep them from sagging in with a tent stake and piece of plastic twine. Guy it out like you would a tent pole or flap.