Is it permissible to work for a foreign company that develops websites containing content that violates religious principles, such as supporting homosexuality, knowing that I am not responsible for the content but for preparing the environment, and would my leaving the job be considered detrimental to the company?

We cannot rule that the questioner's work in the museum or company is forbidden as long as their work is far from direct involvement with objectionable content. What is forbidden in terms of cooperation in sin and aggression is that which is direct or intended. Nevertheless, it is better to seek work completely free from such forbidden things.