How fast was the diffusion of the Gutenberg Press? When were Turks, Slavs, and Arabs able to appropriate the technology? Was there an influx of Latin or German literature in these cultures in the meantime?

I am able to only answer the first part of your question about the diffusion of the printing press. The Guttenberg Press was developed in Mainz around 1440, and by 1500 a more developed version of the press was operating in approximately 150-200 cities across Europe. By 1500 the printing presses had produced more then 20 million volumes of work. In the 16th century this rose to an estimated 200 million volumes. By this time a regular printing press could produce approximately 3600 pages a day.

Sources:

Febvre, Lucien; Martin, Henri-Jean (1976): "The Coming of the Book: The Impact of Printing 1450-1800", London: New Left Books