It could be allergies. It could be a genuine cold/viral infection. It could very well be both together. Anytime children are in an environment with other children who are sick, you can expect your child to get sick as well. Can you take her out of this pre-k and keep her home? Is that an option or is she there because you work? The best place to be is at home, free from viral germs.
I'm surprised the doctor has put her on antibiotics without knowing for sure if it is bacterial or viral. If she indeed has a viral infection, antibiotics will NOT work and you are wasting your money on medicine she doesn't need and flooding her body with medication that she does not need. Now she has a yeast infection. The poor baby is overloaded! The cough could also be viral. Bronchitis is common and also viral. My pediatrician has always said to treat the symptoms...cough medicine when needed otherwise let it run its course. One of my children had a cough for well over two months. I would suggest you get a cool air humidifier in her bedroom for night time. I've used one on all of my children whenever they were sick and it really does help.
Once you child is better, I would definitely get her to see an allergist. I do not know if children are tested the same way as adults but I have allergies, as do my children and yes, allergies ARE inherited! You can bet that whatever mom or dad are allergic to, your children will have the same allergies...sometimes more, sometimes less. I have been tested and it was not painful but again, your daughter is 2 so I do not know how they test children. For adults, it's a matter of sitting still for some time while the allergist places allergens all up and down your arms or back.
Symptoms for allergies are sneezing, runny nose, watery eyes, itchy throat and ears. But some of these symptoms are identical to a cold so you really need to get her healthy first.