How negative is "skip school?" Could a student use it when addressing a teacher, in a neutral sense?

Skipping school is most often something done without permission, and that sense contributes to its rudeness. The Oxford English Dictionary gets at that sense in its phrasing of this definition ("skip, v.1," def. 6b):

6.b. To absent oneself from, stay away from.

"To absent oneself" suggests that the student is usually the one deciding their own absence. Indeed, skipping school, or a phenomenon like "senior skip day" (Wikipedia), is very frequently illicit. Besides the inherent challenge to authority posed by absenting oneself, educators and law enforcement may associate skipping with other negative behaviors:

These teens may also drop old friends and activities, skip school, lose interest in school, receive low grades, sleep in class, lose concentration, and have trouble with memory. (Justice.gov)

If the threat of fines or jail time isn't enough to prompt parents to address truancy, research suggests that teens who regularly skip school are more likely than their peers to drop out of school or experiment with drugs and alcohol. (US News)

Often, the usage occurs when the parents do not know that their child is skipping. That said, sometimes parents may permit skipping school as well, if they see a specific need that goes outside of socially sanctioned cases for absence like a funeral or a religious holiday. One parent explains allowing a "mental health day" in terms of skipping:

In fact, I’ve let my kids, including my preschooler, skip school several times since then, and will continue letting them do this as they get older. Admittedly, my work-from-home job affords me the flexibility to keep them home on a whim. But more importantly, I’ve come to realize something: Kids need mental health days, too. (Today's Parent)

That usage is as close to neutral as it gets, but there's still a slight sense of irregularity. Skipping is unannounced and unapproved by the school. Asking a teacher to skip would be regarded by many as impertinent.

(As StephenS mentioned in comments.) It's not rude as in foul language, but it is also a bit tactless. Use "miss" instead, or be more formal and ask if you can be absent. Not just for school but also for things like, "I'm sorry but I'll have to miss your wedding."

Not an official source, but I found a random internet post that agrees with my gut feeling: skip has a slight implication that you could attend if you wanted to, miss implies that you wanted to attend but are prevented from doing so. Of course, we often soften the truth and use the nice word "miss" when we actually just don't want to go.

News headlines generally don't bother with tact and prefer drama instead so they're not a good source, but in the Thunberg case it's correct. The not going to school was specifically the point of what she did, to draw attention to her protest. She could have protested outside of school hours, but that wouldn't have been as impactful. So she deliberately chose to skip school.