Trisha,
    I agree with the advice you have been given so far, get Alex to a doctor
who is knowledgeable about eating disorders ASAP. Be prepared for the
possibility that she may be extremely resistant to going, patients with
eating disorders are usually very reluctant to acknowledge that they have a
problem. Of course, it may  be something other than an eating disorder, an
experienced doctor is the best chance of sorting it out.
    As to contacting her teacher, I think that could have been handled
differently. Of course, you had to sent some sort of response, but perhaps
something along the lines of
    Thank you for informing me of your concerns about Alex's health. It is
reassuring to know that you care about her so much. I was unaware that her
health was affecting her classwork. Now that I am aware of this, I will
encourage her to seek appropriate medical attention as soon as possible.
would have done the trick, without revealing her emotional problems or self
induced vomiting to the teacher. You say that you did not want her treated
for an acid reflux disorder she does not have, but surely you realize that a
competent doctor bases treatment plans on the results of testing , patient
information, and his or her own clinical assessment, not on what the
patient's teacher (friend, roommate or parent) thinks she has.
    Good luck dealing with this very difficult problem.
Susan