Assignment extensions for no apparent reasons (USA)
The short answer is that opinions on granting extensions vary between institutions and even departments. Some universities view extensions as a way to support student learning. They argue that they want students to be able to get the most out of each assignment and so when the student completes the work is less important than that they do eventually complete it and are able to focus their full attention on it.
Other universities hold the same perspective that you do, that learning to manage deadlines is an important piece of being a student, and that allowing for extensions whenever asked is doing a disservice to students.
The chair of your department or even the other faculty should be able to give you insight on expectations that you should follow and on what is customary. I strongly recommend that you talk to them, since they know your students.
Another option is to pre-emptively institute a limited extension system for all students. The mathematics and computer science courses at my undergraduate college used this tactic to great success. On the first day of the semester, the professor told the students that each of them had a certain number of "late days" (usually 4 for a class with weekly assignments), which would grant a student 24 hours of automatic extension. These could be used at any time on any assignment except for group work or exams. Students did not need to provide a reason for using a late day, although they did need to inform the professor/a TA that they were. There was usually a form online for students to fill out. Students could use them as they wanted- all at once or one at a time. Some professors gave small amounts of extra credit for having late days remaining at the end of the semester, and some allowed students to earn extra late days throughout the semester by doing extra credit work, handing in assignments significantly early, or going to relevant talks, etc.
The benefit of a system like late days is that most students will feel less stressed and more in control of their own time. It thus enables you, as the professor, to hold a quite strict "no other extensions unless absolute emergency" policy. It was well understood that if a student used up all of their late days early on and then got minorly sick or had a job interview or had something else come up, that was just too bad. They would turn in what they had. Further extensions were typically only granted if the student or a family member was in the hospital or if they had formal accommodations through the college itself.
Asking if there is a norm for what is “acceptable” assumes that students care about such norms, but sadly some don’t, and will always try to push the boundaries of what is acceptable. It is up to you to set limits, and this is not especially difficult, and certainly not considered unreasonable in the (U.S., large public university) environment I am familiar with.
I suggest that the next time you start teaching a new class, you include a statement in your course syllabus saying “Deadline extensions for course assignments will not be given except in exceptional, justifiable circumstances”, or something to that effect. Then if students ask, all you need to do is point them to the policy. Problem solved.
As for this semester, assuming you have not had such a written policy that students were expected to have read, it might be reasonable to be a bit more merciful if you are so inclined. But you would still be perfectly within your rights and common sense not to give extensions even then.
Some American students, as members of an individual-oriented consumer culture and economy, will push to get away with as much as they possibly can, in any circumstance, without bounds. The "completion agenda" has given the signal for years that institutions will do everything in their power to guarantee graduation and certification. Especially if students detect that you are unsure or less than confident, they will attempt to get away with more. The US is generally a dog-eat-dog environment.
I sympathize entirely; my first semester had all kinds of missteps on my part like this. The best bet is always say a very clear, confident "No" to any request that you have any doubts about. Maybe say "No" to the first public request no matter the validity to set the tone that you have that in your vocabulary. (E.g.: Once a student asked me, "Can I do X?", and I responded, "Okay, remember, X is prohibited by the syllabus page two...". So she did X anyway and later challenged the penalty mark by saying "But when I asked you said 'okay'!").
Whenever this topic gets brought up, many commenters want to frame it as a debate between (a) good to teach students responsibility vs. (b) good to give students flexibility to learn at any pace. This tends to leave out (c) necessary for the instructor to budget their limited time. Presumably you have some time set aside for grading and assessment and then need to move on to other tasks. (And also (d) good to give speedy feedback on assignments back to the class as a whole.) There's no reason why you should sacrifice your productivity for late student work. They will ask, and there's nothing you can do to stop that. Just say "No".