What is the ruling on money acquired from a PlayStation and computer gaming center project for youth and school students, given that these games contain music and inappropriate scenes? And is embarking on this project unlawful according to Sharia?

Your hesitation in participating in the project is valid, for most of these games contain impermissible elements according to Islamic law. However, if it is possible to provide permissible games, then there is no objection to renting them out. As for the rent obtained from impermissible games, it is unlawful gain.