Low voltage submersible pump safety?

First off the pump itself is a sealed unit (water tight) secondly 12 volts means it is direct current(the 12 volts is from a transformer if it plugs into the wall) and can not shock anything unless it is touching the ground wire or the grounding terminal on the battery or transformer. Thirdly 12 volts is not enough to do anything but scare the cat if it does get shocked.

12 volts is safe, your three prong plug is grounded, just make sure the outlet has three matching holes. The only hazard of low water level is damage to the pump, it's designed to be cooled by the water.

it should be safe as it was desighned for that but you could plug into a gfci outlet that is made for water safety and if it dosent stay under water u could burn out the pumpand ruin it

Anything you plug in that will be submersed in water should be plugged into a GFI. That is the outlet with the two buttons that say TEST and RESET. If you are unsure about anything electrical-DON"T DO IT! it is not worth burning your house down or killing your beloved pet. If the water level is below the pump it could burn out the motor and destroy the unit, or burn your house down. You should call an electrician. A GFI could easily be added to any circuit and give you peace of mind.

1) Voltage doesn't kill. Amperage does. As little as a few milliamps. If you are solidly grounded, can't get loose and have a weak heart. The amperage interferes with the signal to the heart and can cause it to flutter. Any electrical device in your home has that potential. The good news is that you are not likely to be unable to get loose. Same with your cat. This is why animals nearly always survive chewing through electrical cords. The initial shock cause muscles to contract and in so doing we/they break contact. That's not to say it doesn't hurt!
2) Low water level will result in failure of the pump. It burns up internally. Should not result in current escaping to ground. ie being an electrical shock hazard.
3) I have no idea what they mean by "always run under direct supervision". Try contacting them for an explanation.
4) Yes, the unit itself is of the grounded type. Unfortunately, just because the receptacle you are using has 3 prongs doesn't mean it is properly grounded. In older homes that didn't have a ground it is not uncommon to find the 2 prong outlets have been replaced with 3 prong type without the addition of a grounding means. If your home is say less than 25 years old, they most likely are grounded. For about $6 you can get a 3 prong outlet tester and plug it in to the receptacle. It will tell you if it is or isn't grounded. (spend the extra #3 and get one that also has the push button on it. That feature allows you to also test GFI receptacles and GFCI breakers.)
Replacing the existing receptacle with a GFI isn't all that hard. The key is identifying the incoming power (Line) and putting those wires in the correct location. If additional wires are present in the box you have the option of putting them on the Load side terminals of the GFI and thus having GFI protection on outlets "downstream". Or putting them on the Line side terminals along with the incoming hot. This way should the GFI trip nothing else (downstream) looses power.
Take a good look at the paper work that came with. Do you see the UL label listed anywhere? If not then I would be concerned. If it is present, you should be just fine.

Low Voltage Submersible Pump

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RE:
Low voltage submersible pump safety?
I've bought a pet fountain with a submersible pump which operates on a 12 volt adapter. Soon I have second thought since I know nothing about electricity or safety of electrical appliances. Would anyone please advise:

1. In case there's any power leakage into the water, is my cat safe...

When in doubt don't. As far as it's safe for any thing from an electric shock. It is not possible. It takes 2 parts of the circuit before any shock can happen. Unless the pet is standing on a conductive surface the pump can not provide a shock.