Is it appropriate to try to research the details of a companies hiring
process while working there as an intern?
Unless prohibited by your internship agreement, it's perfectly appropriate to read anything accessible to all interns, and to learn whatever you can from your readings.
Is there a good way to approach this to be able to better interview
for a potential FT position?
During your internship, you should learn everything you can about the company, the people, the jobs. Read, network with others, ask lots of questions, work hard.
When interviewing interns for full-time positions, hiring managers expect them to know more about the internals of the company, and to have a leg up on applicants who haven't been interns there.
Be aware that the interview questions document you read may not apply when interviewing interns. The interviewer may ask different questions, since you are more of a "known quantity" and have internal knowledge that others wouldn't have. Still, it shouldn't hurt to read and understand it.
(Note that it might not be acceptable to disclose any of this information outside of the company.)