Is it considered a sin to wear headphones while working, and is it permissible to sit down to rest when unable to stand throughout the entire shift, knowing that the work conditions require not wearing headphones and standing?

The condition of not wearing headphones at work is valid and binding, for the benefit of the work, and Muslims are bound by their conditions except for a condition that makes lawful unlawful or unlawful lawful. As for the condition of not sitting, if the nature of the work requires standing, then it is valid and binding. If it does not require that, then it is an arbitrary condition, and the worker is not obligated to abide by it. If you find another job that does not impose arbitrary conditions, that would be better.