I don't understand what she means by having to feed her brother when he is sick. Is that because he's fussy and so the novelty of having her feed him takes his mind off his troubles? That may make some sense, but do you think she feels that she's getting sick because she's feeding him? Then you and her father have to disavow her of that notion. Yes, exposure to other children can help germs transfer, but it's no different from her being in school with other kids, whether or not they have siblings. So please try to reassure her that she needs to take the same precautions everyone else takes - sneeze into the elbow, wash hands with regular lather (not antibacterial soap!) after being with others and after using the bathroom. I'm not sure how old she is but have her sing the ABCs while rubbing her hands with soap, and make sure she's not doing what a lot of kids do, which is to put a little soap on and then immediately put her hands under the faucet to wash the soap away.
It doesn't help that her father gets defensive instead of partnering with you as co-parent, but be sure that he doesn't think you are blaming him for exposing her to germs. She's getting those everywhere - grocery store, after school programs, the school bus, the kid next to her in gym, etc.
I'm confused by the number of days - she gets 9 days a semester, which is 18 absences for the year. She's had 4 days for fever, and a few days for dizzy spells every once in a while, so how is she nearing 18? I'm not clear what's causing the dizzy spells - and you don't need to share that - but it's clear that you've discussed with the doctor. If extra hydration would help those (and it would certainly help the colds/flus), then the same remedy would address multiple issues.
I think the number of absences in most schools is "unexcused" - so if she has a doctor's note, it would be an "excused' absence. My son had mono and was out for quite a while, but he had a doctor's note and so there was never any issue with the school. If a child is out a lot, they provide tutoring or extra help.
The school should be really used to kids with separated/divorced parents, so make sure her father automatically gets all the school communications (general as well as those specific to her). If they come from the school and not from you, maybe he will be more proactive.
I have step kids and I do urge you not to get all your info from a child who is going back and forth. It's so incredibly difficult to avoid that, I know, but from my experience it really backfires.
You also have to let your child take some responsibility for herself - again, it's hard because you don't mention her age - but consistency of bedtime and nutrition can help a lot. Is there a way you and her father can co-parent with that? Is there any possibility that allergies are involved? Being in enclosed houses during the winter increases the number of germs people are exposed to, as well as allergens. That's not something to blame on this house or that house - it's also her friends' houses and her school and her after-school activities.
There's also a lot you can do with immune system support, if you and her father can agree on it. I work in food science and we see a tremendous decrease in allergies and sick days when kids, and the adults in their word, are doing this.
Good luck, and I hope the end of the school year approaching means that this issue will be put to rest just by the calendar!