Is one fill enough for a proper clean?
Yes, absolutely. One fill is more than enough.
Is it a good strong stream of water?
The water jet is strong enough and is comparable to a stationary bidet or jey spray.
Does it become dirty after using it?
No, it does not become dirty since it doesn't touch anything on your body. It is not intended for contact and is engineered to hit the right spot with very little splashing.
Do I need to install anything?
No, you do not need to install anything.
Is it quiet enough to be used in a public restroom?
It's not absolutely silent. People nearby will be able to hear it, but it's quiet enough that it's not generally disturbing.
Does it warm the water?
No, it doesn't warm the water. You can fill it with tap water at the temperature that's just right for you.
Because it can carry liquid, does this mean it is prohibited for carry-on use on air planes?
It is fine for carry-on use on air planes. The airport security restriction is for the liquid itself not bottles or devices that carry liquid.