What is the ruling on downloading or playing directly from gaming websites that contain music, with the sound muted, or bad advertisements, with them blocked? And does their popularity increase thereby and lead to sin?

There is no need to download games that contain prohibited music, even if the sound is muted. Fatwa no. 130204 guides and recommends avoiding this, but does not definitively prohibit it. The correct view is that playing them with the sound muted does not appear to be forbidden if they are free from other Sharia prohibitions, because the reason for the prohibition is listening to music, and the ruling ceases when the reason ceases. There is no difference between playing them on the website or downloading them from it, and downloading them is not considered promoting the website's content.