Is touching a ground wire dangerous?

Just to reassure the OP, I have actually seen a little girl get zapped and end up in intensive care, by taking a drink from an outside faucet hooked up to a plastic pipe. Her father had just had a new electric oven installed with a defect. It did not disrupt circuit protection (30mA here) for unknown reasons, and for other unknown reasons the water mains were sufficiently separated from sufficient grounding to transmit a 180V shock. The kid is all right, the problem was solved without finding cause, but the underlying stupidity could have happened to just about anyone.
Just try not to put yourself in “wire-like” situations, even between grounds

And the ground lead in the device’s power cord is, in turn, connected to the device’s case. There’s no point in putting a three-prong plug on an appliance unless you’re going to connect the other end of it to the appliance.

Harmonious Discord is probably going to accuse me of fear mongering again, but “enough to scare you” is enough to kill you. It’s not terribly likely to kill you, but it can kill you.

Electricity kills you generally in one of two ways. Low level shocks (the kind you would get from a misbehaving kitchen sink) have a chance of interfering with your heartbeat, and can throw your heart into fibrillation. Your heart has kind of a funny design that if you get it into this state, it will happily stay there until something (like someone with a portable defibrillator) comes along and gets it out of it. It takes a surprisingly small amount of current to throw your heart into fibrillation, but it is very hit and miss. Your heart is much more sensitive during certain times of its cycle than others to this sort of thing, so most of the time a person just gets scared and that’s the end of it. Once in a while, though, you catch the heart just right in its rhythm and then you’re screwed.

The funny thing is that higher level shocks are less likely to kill you. If you get more energy, instead of going into fibrillation, the heart muscles just all contract at once. Your heart isn’t pumping blood, so if the source of the electricity isn’t removed you can still die, but generally once the source of electricity is removed the heart will go back into a normal rhythm.

At even higher levels of current you start running into the second way that electricity kills you, which is that it literally cooks you to death. This is how the electric chair and lightning bolts kill you, and this is a lot less hit or miss. No one ever walks away from the electric chair, and those fortunate enough to survive lightning strikes often have very long recoveries to heal the burned tissue.

For low level shocks to harm you, the path of the current has to be through your chest, which it probably was with your misbehaving sink. Just grabbing both prongs of a plug and shoving them in the outlet won’t kill you though, because the current only goes through your thumb, hand, and finger. You may end up with nerve damage and maybe a burn, but you won’t die. It’s very easy to get current to go through your chest, though. Many folks don’t realize how much of their house is electrically grounded. All of your water pipes are grounded. Even metal window screens can ground through aluminum siding (I found that one out the hard way). Touch something electrically hot and something grounded at the same time, and ZAP.

It used to be that in old houses the electrical system had to be grounded through the cold water pipe. Over the years, they found out that this wasn’t such a good idea. Corrosion could easily cause a bad ground (a possibility for your sink), and you could completely ruin the home’s grounding system by replacing a section of pipe with PVC. These days, a separate dedicated ground rod is required. But you don’t want all of your water pipes and the metal parts of your house to float in voltage, because if there’s some sort of electrical fault they could all end up electrically hot. So these days, even though we don’t ground through water pipes, all of your water pipes and such are required to be connected to electrical ground. If you replace a section of pipe with PVC, you should run an electrical ground to the metal pipe on the other side of the PVC so that you don’t ever run into a “hot sink” situation.

Personally, I’m not worried about myself. I installed the adapter, and ran a wire to a water pipe in my old house to ground my computer outlet.

I’m worried about my dog. Right now, my main computer outlet is surrounded by furniture and a garbage can to keep my dog out. My dog also sheds constantly. I’m worried that her hairs might fall, touch that ground, spark, and possibly smolder or even set fire to the carpet.

Of course it’s dangerous!
That little loop is there to bleed off dangerous current. That’s what it does.
If you are running a machine that has a third prong, like a commercial vacuum, it has a motor that is building up a static charge that has to be drained. And of course because it has moving parts when something goes wrong the housing will be electrified, and the prong is to bleed it off.
Bypassing the ground by using “cheaters” is cheating yourself of safety rules established in the 1920s because too many people were being electrocuted. It doesn’t take very much current to throw your heart off rhythm and cause a heart attack.

in a grounded appliance the grounding wire carries no current and is at ground potential (for all practical purposes) during normal operations. it is only when a fault occurs (the hot wire comes in contact with the device housing) does it carry current creating a short circuit and instantly tripping the fuse/breaker.

that was good protection and later Ground Fault circuitry offered even better protection.