I just completed my MFA in Film this August, 2017. Now I am freelancing and not associated with an academic institution. May I still use my graduate university's letterhead when applying to faculty positions? I don't have any other nice looking letterhead with a logo, unless I make it up.
No you may not. The use of an official letterhead implies not only a current affiliation with the university, but it some countries and contexts also implies that the correspondence is for official purposes. Each university has it's own rules and policies, but I would be shocked if your university would consider this acceptable, and both universities that I have attended have specific guidance to not do this. In general, you do not have the right to use that letterhead simply because you attended the university, and using it in that fashion is inappropriate because you are representing yourself as having a professional position that you don't have.
Comment posted as answer: I'd encourage you to contact your school, especially the career services office if there is one. When reviewing job application letters during hiring, I often see some institute-issued alumni letterheads. Maybe your university/institute has something similar. It's in their interest to see you become employed, so ask them how they can help you to best showcase your academic background. The worst you can get is a "no."
In short, you are using a letterhead if you are "speaking for" the institution whose name is on the letter head. This would, for example, be the case if you're an administrator writing to one of the suppliers of the school cafeteria, or if you're a professor writing a letter of recommendation for a former student.
In your case, you are writing for yourself. You do not have an official role in the university, nor are you speaking on its behalf. Don't use the university's letterhead.
My recommendation would be to create your own letterhead. As a freelancer, you need your own brand identity anyway. And, working in a creative field, I imagine that anyone you contact would expect you to have the skills to make a letterhead.
The letterhead of the university will be copyrighted and will not have your contact information in. It's very unlikely that you will permission to use it, or that this will be granted even if you ask for it.
As a further example, in universities where I have worked, letterheads have been carefully protected even for staff. Only a small group of staff would have access to them and the would record how they were used. For instance, if I needed to write a reference letter for a student, I would not be able to directly print it onto headed paper. This would need to go through an assigned administrator.
Now, in practice, of course I could very easily have scanned in the end result and created my own letterhead, but that just wouldn't have been acceptable. If a student were to have done this, the matter would not have been taken lightly.
The closest suggestion I can make, if you do want to include a visual representation of the university on your letter, would be to see if you can get permission to include their logo. This will be less controversial than the full letterhead, but it might also be protected.