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. How can a TA balance empathy and deferring to the syllabus in teaching responsibilities?