The used to be defined as one kg of water, but it is not an official SI unit (it is part of the SI accepted metric system, which does not require such rigorous definitions).
"water at the melting point of ice" from an old definition
Exactly, so you need to add that or it won't be true most of the time.
Inofficially, it has been. Technically, it's not quite true. Water has a density of 999.97 kg/m³.
That's unfortunate. How did it come to be that way?
Water is too hard to keep constant, and too hard to reproduce. Keeping it exactly in 1dm3 is tricky, for example. Not for "regular" use, with a few significant digits of course; but when you're defining fundamental physical constants, atoms count.
That depends on the temperature ...
At 0C
Not really. Because of temperature, usitipe composition, etc, the weight of a liter of water is not precise enough for physicists.
One kg of water is definitely one kg.
The litre is defined as 1dm^3, the mass-based definition was dropped in '64.