What is the ruling on watching or downloading free programs from the internet—such as cartoons, movies, and games—without knowing if they are official, and what must be done to absolve oneself of the rights of their owners?

It is permissible to watch, download, and install free programs, movies, and games that you do not know to have copyrighted material, provided they are free from Sharia prohibitions. You do not owe any right to their owners, and you are not obliged to research and verify whether their download is authorized.