A necessary caveat, that many of these are opinion questions and thus may vary from professor to professor. I'll try to go through each of your questions but some may overlap
What are some factors that a student should consider when asking for help with course content and assignments?
Is it in the syllabus?
Was it covered in class/the textbook and would a revision of that help me understand?
Is this something I should already know? ie. A foundational concept that is lower-level than the course itself
Other than that though, I think students shouldn't be discouraged from asking questions. I do think there's an important factor that isn't about the type of question itself:
I get too many questions via email that need to be discussed in person (eg. "I don't understand this concept,") and there are many questions that the whole class would benefit from and thus would be best asked in tutorial.
Professors, barring any questions that can be answered in your syllabus, are there any stupid or inappropriate types of questions a student should avoid asking or would raise some red flags about the student? What is your response to students who ask bad questions or too many questions?
Your example is pretty much my answer: questions that demonstrate an inability to meet the requirements of the course. Like you said, a math student not understanding base concepts or how to use a calculator. It's not necessarily a red flag that the student is a bad student, I feel I should clarify, but it is a red flag that the student is going to struggle through the course. If they're asking questions about basic things, how can I expect them to conquer higher level material?
Too many questions, I'd probably whine about to a friend but I'd answer all of them. Some people are inquisitive. Bad questions, I'd do my best and direct them to other resources.
How about when a student, who has been a slacker all semester, suddenly starts requesting help?
Wary, mainly. Why are they asking for help now? Suspicious that this is going to be a burst of interest that will rapidly die out and the student will return to being a slacker. I'll still help them, of course, but I'm more likely to direct them towards materials they previously slacked on. Some students do genuinely turn it around part way through a semester, most don't. Most start asking questions, realise this won't get them the grade they wanted, and give up again.
Is there balance a student should consider between asking for help and helping themselves when they are struggling through a course?
This is a tough one. I think the student should do the best they can to help themselves, but obviously there's a limit. If they're struggling and not getting anywhere on their own, they need to ask for help. But I would encourage all students to try helping themselves before rushing to ask a professor for help.
At what level of higher education, if ever, does asking questions start to look bad for students?
I don't think ever, unless it's the previously discussed foundational questions: a PhD student asking how to conjugate a verb in the studied language, or how to use a pipette are going to reflect very poorly on said student. Or, any question where the answer was easily found on their own, and the question didn't need to be asked. But otherwise, no, I don't think asking for help looks bad.