Yes, absolutely. One fill is more than enough.
The water jet is strong enough and is comparable to a stationary bidet or water faucet.
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.
No, you do not need to install anything.
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? Or do I have to wait by the sink for warm water and fill it up each time before I use it?
No, it doesn't warm the water but room temperature water is quite comfortable, even refreshing.
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.