Coming late to the thread, but thought I'd weigh in, too. Lots of thoughts going through my head reading here.
First, you have to remember that the throat and stomach are part of a bigger
system, i.e. the gastro
intestinal/GI tract. Nowhere in this thread did I see anyone ask about, or discuss, bowel movements! That will be one of the first questions a health care provider will ask with a presentation of recurrent vomiting. Kids' diets are horrible these days. They're eating very little fiber and many are experiencing chronic constipation. Part of a healthy lifestyle is daily bowel movements that are relatively soft and formed. It's not unusual to see kids who haven't gone in a week! And there can be vomiting associated with chronic constipation (even that that may not be week long, but even a couple of days), which often also goes along with an unhealthy diet. So Mom, first thing to look at is, how is her diet and how is her GI tract doing, as a whole?
Second, MOM INSTINCTS!! Mom's feeling like the Slime could be a problem. Well then, it probably is!! Stop it and see what happens. Always pay attention to your instincts!!
With the addition of the low back pain symptoms, I think posters who mentioned hormonal changes are probably spot on. It's happening earlier today than when we were growing up. And you have to remember, it's not something that happens overnight, it's a
process. It could very well be that her hormones are beginning to change, and because it's something her system isn't used to, it's causing some vomiting and low back discomfort. Something to just keep an eye on and think about. Probably not a bad time to start educating yourself and your DD on this process if you haven't already. A little ibuprofen can help the back pain, but taken with food, and of course, be careful since it can be upsetting to the stomach.
Anxiety. Yes! That was one of my first thoughts when reading the OP. Why? Because with a stomach bug, you don't usually have continued vomiting once that part "passes". Often GI bugs start with vomiting, then turn to diarrhea as the move down the tract. Here, we had vomiting, then more vomiting a day or two later, with no mention of diarrhea. And no other reported symptoms. Unlikely to be a bug. (Or any of the other conditions mentioned, since there were no other symptoms, either, which would be typical. Yes, yes, we all know cases where it's happened, but I'm talking typical here.)
Anxiety can definitely present as stomach upset. OP, I am so sorry you lost your sister like that. That must've been awfully hard. And I can understand why you'd have anxiety about your own child's health. But you must be careful about transferring your own health-anxiety on to your daughter. It's easy for kids to pick up on our anxiety It starts when they're babies and continues on, so by age 9 there has already been a lot of exposure, and it's possible it's already become a way of life for her, too. So this is something you can work on.
And please know I'm not judging, just educating. Believe me, as a parent myself who sees a lot of bad things medically, it is sometimes hard for me, too. I had a friend die at age 18 of a bone tumor in his leg. Imagine my horror one day day when my son was young to see a big lump on his shin out of nowhere! I called our pediatrician and she was so calming, she said she highly doubted he had a bone tumor, and that I should watch him to see what was going on in his environment to see what was causing the lump. So I did, for a few days. Then one day I watched him climbing the wooden ladder on his bunk bed. When he got to the top, he put his shin right on the wood as he launched himself on to the top bunk. Bingo! We padded the top of the ladder and the lump went away. I could've rushed off to ERs and put my son through testing and whatnot had I been really anxious, but I trusted my pediatrician (Man, I'm going to miss her!) and looked for a more logical answer. I think you should do the same with the Slime.
My own DD is 20 now, but we were always having to figure out weird things with her, and that continues to this day! Recently she kept telling me her throat was bothering her, but it didn't happen until her college nursing group was doing clinical exams, and it started when someone examined her thyroid. I kind of didn't put a lot of stock into it since she was otherwise relatively well, but it continued to bother her so she went to the doctor's. Well they couldn't find anything wrong so they referred her to an ENT. She had a scope and turns out she did have a pocket of infection! As well as reflux. So she did a course of antibiotics and a PPI and she's better. Although, literallly, as I'm writing this, she just texted me to say she has a swollen lymph node in her neck now.

When I'm done here, I'll be writing her back to say it's not unexpected with a recent infection, but we'll continue to watch it. (I, myself have had a biopsy of a neck node which we still have no idea why it was swollen, but all infections that happen in the head in winter pass through neck nodes so it's not surprising they get swollen from time to time.)
I think we all worry about our kids, even when they get older, but we have to balance it with helping them develop a sense of control over their minds and bodies, as well, and teach them how to think logically about things that relate to their own well-being. Good luck, OP. (IMO, the Slime would be GONE!)