Yes Sinus Infections can cause bad headaches.
It is an infection. In the sinus cavity.
An infection.
And, little kids, do not know how, to spit out their mucous. And yes, it can cause vomiting because of the mucus quantity being swallowed and due to post nasal dripping and the amount of mucus.
Antibiotics, serves to, eradicate the infection.
It is not a headache pain reliever.
What is she eating and drinking?
Junk food or healthy food?
Sinus Infections or head colds, can also affect the rest of the body.
It is, an infection. And coupled with Fever, the whole body is working very hard at fighting it.
I had a head cold once. And then, it progressed. Into, a bronchial infection and almost Pneumonia. Why? Because- any mucus that oozes down the throat (ie: post nasal drip), can sometimes, go into the lungs. Then, the lungs gets infected. Via the mucus. Then at that point, I had to have antibiotics. To combat, the whole infection system.
You CAN take your daughter back to the Doctor, for a follow up. And just to make sure.
That is what I would do, AND tell the Doc about her weight loss.