All symptoms of pneumonia except fever. Could it still be pneumonia? Chest pain, bad cough, trouble catching breath, slight wheezing, etc.

Can be pneumonia? Symptoms are consistent with bronchial process, can be asthma or an allergic reaction. Best to discuss with your doctor.
Yes. Yes, you can have a serious infection without fever. However, it is not likely that you have bacterial pneumonia without a fever. Many of the cases of community acquired pneumonia are viral disease. In fact, in several studies, viral causes for pneumonia were more common that bacterial causes. Viral pneumonia can be severe, though. If have severe symptoms, you should be seen by a doctor.