Balancing empathy and deferring to the syllabus in teaching responsibilities
It sounds to me like you are allowing yourself to be emotionally manipulated by your students, particularly the one (who sounds super obnoxious) who told you about the promise for a luxury car from their parents. It’s indeed important to have empathy, and I find your conscious insistence on practicing empathy every day incredibly noble and admirable — if only more people did that the world would be a better place — but at the same time, sometimes a person has a job to do and needs to put the larger interests of the mission he or she serves ahead of the emotional need to “be nice”. Indeed, being nice to people who are undeserving is often harmful to those people. You would not be doing any favors to the immature, manipulative student who wants to get a luxury car by giving them unfairly favorable treatment out of pity or “empathy”. (The student with medical problems is a completely different story, where the request sounds a lot more reasonable.) 

In any case, empathy is about how you relate to people and not about the decision you make on their petition. Even on occasions when I’ve had to turn down a desperate student’s request for a grade change, and knowing that that’s the only decision I could possibly make in good conscience, I would like to think that I explained my decision to the crying student with empathy - I certainly tried my best, even if the students in that moment sometimes only seemed to care about having their request denied and might well have thought me to be a cruel, heartless person.

So what is a good way in practice to solve the dilemma? The answer you quoted in your post has it exactly right. You give an extension to the people who have a legitimate reason to need it, and refuse it to those who don’t. That’s all there is to it. “But what is 'legitimate'?” Exactly what you judge to be legitimate — after first pausing to take ten deep breaths, counting down to zero from one hundred, writing an email draft answering the request and saving it in your drafts folder, going on a walk or a coffee break, and finally coming back and looking at the question with fresh eyes. If at that point you still think the need is legitimate and can say with certainty, or least with reasonable confidence, that you are not falling victim to emotional manipulation from an unscrupulous student, by all means grant the request. 

And whether or not you‘ve decided to grant the request at that point, be content to know that you have been true to your principles of practicing empathy, and that you are doing your small part to help make the world a better place.