Is it possible to view how many times any given game on Steam was purchased or downloaded? Steam & Game Stats only seems to list the top 100 games (but doesn't include purchases) - is there a way to get this same information for any given title on Steam?
Valve doesn't release the sales information of games on Steam. They do share their statistics with the developers of the games, but the developer decides whether to share this information further. Source
It may be possible to deduce some statistics from the publicly available information, but I have yet to see a public API that gives access to more in-depth information than that available on the Steam & Game Stats page.
Steam Spy is a website allows you to search for a specific game, to get estimated sales numbers and hours played.
Although Valve doesn't make such data public, it is possible to estimate it, based on what data is public. Steam Spy does exactly this by random sampling of public user profiles, which list what games that user owns and how many hours they've played. They then multiply by the total number of Steam users. You can read in full about the methods they use, over here.