Personally, I have a running list. Every time I come into a situation, regardless of where I'm working, and there's a question of "who is really responsible for this?" I add it to my list. At the top of every new job, I sit with the PM (or other person in that general job-title-area as discussed above) and go over my entire list. At one job I actually had separate sit downs with the PM, the Company Manager and the Casting Director - just so I could get all of my answers.
After I finished my gig at that company, I updated my question list to be "my task list" for that company. I then generalized the question list to prepare for the next theater - I asked all my questions, updated the list even more, then saved again. It's been extremely helpful for me to always have this long list of questions to be prepared with.
And now, after you've gone through this experience, your list can really start to take shape. Write down all questions you've had and their particular answers for this show. For the next show, sit with the TD again and review the questions for that particular show - responsibilities may have changed. And granted, depending on the job-title mix that the theater has at any particular time, responsibilities may shift show to show. (How about working without a Company Manager and an Artistic Director? Going through that myself right now, and responsibilities are shifting to lots of people, while new jobs are created. I've been with my company for 9 seasons and we're still figuring it out. We just ask a lot of questions of each other to figure out who really does what.)
While the handbook sounds great, you never know when a company may decide to reorganize, or companies may discover than an assistant handles a task better than the boss, and responsibilities are constantly shifted. Go show by show, keep your list of "what it was last time" and always ask questions. If you don't ask, you're never going to know.