Questions of similar topics have been asked here for the situation of instructing a course and here and here for the situation of 'no late work' policies. Additional questions about balancing teaching responsibilities here.
I am a new graduate student, and part of my funding comes from TA responsibilities. For the current particular course, I am responsible for teaching lab sessions, grading homework, responding to student emails/questions, and other common duties. So far, I have not had too much trouble balancing the responsibilities of being a teaching assistant with other commitments.
Homework assignments are always released two weeks ahead of time and are due 9am on Monday mornings. For whatever reason - length of the homework, difficulty of a specific problem, etc. - this homework has caused several last-minute (example: midnight last night, 2am, and 6am) requests for extensions.
The situation of late homework requests is confounding to my daily practice of empathy. I feel like I am at a split in the road or forced to choose which devil to listen to. For example, one of the requests is legitimate from a student who has been experiencing medical issues the entire semester. Another is from a student who has explicitly articulated they need a good grade to get a luxury car from their parents. I have a personal rule to actively practice empathy in my day-to-day life, but my rule completely falls apart in this situation.
In @Paul Hiemstra's answer on another post, the use suggested:
If people have a legitimate reason for wanting an extension of the deadline, I would simply give them the extension. If they cannot provide a good reason, i.e. they simply procrastinated too much, they have to take responsibility for their behavior, and they do not get the extension.
But what is 'legitimate'? Additionally, I am concerned that individually evaluating the reasoning of each student - explicitly stated or not - may lead to personal inconsistencies in who I give an extension to and eventually consume large amounts of time.
I've spoken with the course instructor, but he has left it up to me. My university and department have no centralized policy. How can a TA balance empathy and deferring to the syllabus in teaching responsibilities?