K.N.
Based on her history of chronic infections, it doesn't sound like teething is the root cause of her ailments. She very well might be teething... However I don't think that would cause her reoccurring infections.
I once lamented to our pediatrician that my daughter was born congested! Around 10 months, my daughter got her first sinus infection... Between then and 23 months, no kidding, she had chronic sinus infections every 6-8-10 weeks... Ear tubes at 15 months (to no avail, because sinus infections persisted), ER visits due to developing allergies to repetitive use of antibiotics, pink eye, multiple rounds of sinus x-rays, diagnostics to rule out a heart condition... almost did the test for CF however she was diagnosed with an allergy to cows milk/dairy right before we went down the CF path. I wish I had clued in to the fact that an allergy to cows milk proteins could cause congestion and sinus issues. That would have saved me so much anxiety and stress, not to mention keeping her from developing antibiotic allergies!
Since getting my daughter totally off cows milk, she has been antibiotic free. I strongly suggest that you take her to a pediatric allergist and test to see if she has any allergies before you inadvertently repeat my mistakes.