Can I pursue a project outside of my major department or with a faculty member I haven't had in class?

Yes, for sure. If you are a learner, you can accept challenges to meet your goals.

yes, you can collaborate. the fellow may need something from you and you need something from him. mutually benefited. sharing knowledge and making significant impact. but seldom it works in asia and middle east. here each one tries to steel from the other. failed miserably in africa and india. collaborative reseach centers are shutting down one by one due to conflict. now it is like purchasing science as i see in UAE. several universities in UAE are purcahsing scientist from all over the world so that they can steel the information, not share the knowledge. to do something, you must be expert in something so that the other expert can collaborate