Gender Assigned at Birth: Male - Female (Correct me if I'm wrong but biological sex and what is assigned at birth are the same thing)
Gender Identity: Male - Female - Transgender - Non-Binary
Socially Presenting as: Male - Female - Non-Binary
So transgender covers the spectrum of terminology dealing with most gender queer individuals which could help limit your responses while providing that option for a wide variety of people.
Adding the socially presenting as would help determine if they're not able to present in the way they feel they should be. Since your study is about figuring out if someone identifies differently then they are able to appear is the main objective?
But I agree, these questions are extremely personal for a class paper of any kind. I'm not completely familiar with the terminology though either. I prefer being referred to as female or transgender myself, since it covers a spectrum instead of specifically labels, and from my experience trans-people have many personal opinions on how to refer to themselves in this way.
PS: My own personal struggle is the similar with Stacy. I've always felt feminine, my personality, the way I speak and how I act has all aligned with that so I've been a #1 target for a lot of abuse. People (mostly men) see a 'soft' guy and see a target for bullying and whatever else they can come up with to boost their own egos. Then there is the aspect of overwhelming anxiety when doing things like applying for jobs and going on interviews.
Using myself as an example, I'm trans 100% of the time in life... EXCEPT for work because I cant get a job with MALE stamped on my application, having a male name, handing over male birth certificate and license, then sounding, behaving and looking feminine. This is extremely nerve wrecking for me personally. The looks I get while standing there cashiering for 8 hours straight.. Ugh. I'd like to think if all trans people had the resources and community around them to support their transition those numbers would be MUCH MUCH lower. instead we are faced with crippling poverty and discrimination in the work place making it difficult for us to provide cash flow for ourselves. I mean, how should someone feel when they know that there are surgeries and resources available to help them with their dysphoria but.... you can't afford them so you don't get them, ever.
Being able to quantify all of that into a survey is very difficult.