Ok, first of all calm down. There is no need to worry. I will try to answer you as best I can from my experience with this. It seems like you are in good hands as far as the doctors go. I don't know if you have ever had bronchitis, but when you do have it, it hurts....I had it as a young girl, and every breath was extremely painful. This gives us the tendency to breathe more quickly in order to get more oxygen to our lungs, instead of taking normal breaths which are deeper and more painful.
As long as the baby is pink, then she is getting enough oxygen. The only thing that worries me is the fact that a high rate of breathing is exhausting after a while. It never hurts to get a second opinion on what do to about that... your babies might need quite a bit more sleep than usual because of the strain of breathing. I don't think they will die.
The reason why an antibiotic was not prescribed, I don't know. Sometimes it is better not to have them...some bronchitis comes from viruses which will not be effected by antibiotics. The inhaler is good. Did they give you a vapor machine? Usually comes with an air compressor and a mouth or nose piece that can clear up the lungs more quickly, and depending on the drug used, can also help babies a great deal.
If the baby has a fever, his breathing is going to change too, so this breathing may be unrelated to how his lungs feel. As always, go ahead and get a second professional opinion of a pediatrician if you don't think you are getting all the help and advice you need.
I hope this helps and that your babies are better soon!