If it is possible, how would you estimate a compound's water solubility from data like its partition coefficient, $ \log P $, its dissociation constant, $pK_a $, its molar mass, etc.?
It could certainly be estimated from those properties, the structure, etc. A lot of companies have developed proprietary algorithms for things like that: solubility, diffusion coefficient, polarity.... (For example of a company, see "ChemAxon")
The water solubility for many chemicals is already known, though. Search for a chemical ChemSpider, and look under the "Properties" tag.
They've even got an API, in case you're interested in using water solubility in an app.