Is it normal for my baby to have a runny nose when teething. She is 10 months old now?

Normal. While teething is not the cause of runny nose it is coincidental. It is that age when they are teething and also catching minor colds. If there is no other symptom i would just watch.
Yes. Slight runny nose, low grade fever and crankiness are often signs of teething. If she developes an extreme runny nose and/or heavy congestion then she should be evaluated by her pediatrician to make sure she doesn't also have a cold as well.
Yes. Think about it...When children are teething, teeth are literally cutting through the gums and the mouth is full of bacteria. Many children then get mild symptoms like runny nose or cough and the immune system is doing its job. No worries, she will be fine. I do like hyland's teething tablets or humphries #3 for teething.
Likely unrelated. An infants nose runs when they are exposed to airborne irritants, like dust, fragrances, cooking fumes, etc. It will also run with minor colds. They also cut teeth about 1/mo starting at 6 mo. So at any given time a runny nose in a 10 mo cutting teeth would be the norm rather than the exception.