Is it said in the game that Project Purity boils the water to purify it?
It isn't, as far as I know, I don't believe it's ever explained exactly how it's done.
But in any case, while using soil for water filtration might do what Project Purity was meant to, it would probably be rather difficult to do it on the scale that the project requires, and in any case, Project Purity was mostly a plot device used so that Bethesda could tell the story they wanted, I think, it gave James a reason to leave the fault and gave you something to fight for after finding him. In the end, as has been said, Fallout is a little light on the "science" part of science fiction. There are, of course, science-fiction settings that are even softer than Fallout, but it's still far from what I'd call "hard science fiction", so the real problem we must be concerned with is not whether any addition to the world is consistent with real world science, but whether it is consistent with the rest of Fallout.
Nope its just irrelelevent The vaults have clean water because they have some computer controlled distillation process. Thats what your quest in FO1 is about. The water purification chip fails and you have to go find a new one. The process is very ineffecient and complex and is really only suitable for supporting small numbers of people.
Radiation wouldn't be a problem, but realistically, sanitation would, however, that doesn't play into Fallout 3.
This is also the reason for the mutated species. Check out The Info? as well as reading the Fallout Bible at the Wikia site if you're truly interested in why things are as they are in the Falloutverse. It's not all radiation, but a combination of Radiation and low-level FEV exposure. I'm sure that Limit 115 also has something to do with it. There's much more than Radiation at work in the ground/air/DNA of the Falloutverse..
Still, even with FEV involved, it's still unrealistic, because I'm fairly certain no virus could completely rework the DNA structure of an entire person, and if it did, it still probably wouldn't show the changes until the next generation, and certainly not almost immediately as seems to be the case with super mutants.
The desertification is in the world of Fallout caused in large scale by the exposure to FEV which was released cause of a nuclear hit.
It seems unlikely that a virus that is supposed to cause near instant evolution (how that works I don't know.) will induce desertification, it seems much more believable to just assume that the blast and radiation killed the plants and caused the state of the soil we see now, though that doesn't account for the utter absence of rain.