Can you have pneumonia even if you don't have the coughs?

Yes. "walking pneumonia" is an indolent, more subtle form of pneumonia that can occur. It can cause some mild cough, fever, and fatigue, but it doesn't present with terrible cough on occasion. A chest x-ray would reveal the pneumonia, and specific antibiotics are designed to treat the bacteria commonly associated with that type of pneumonia.
Yes... Pneumonia is an infection of the lung parenchyma and there is usually an associated cough but there doesn't have to be. Other signs and symptoms include fever, shortness of breath, body aches, etc. The diagnosis is confirmed with a chest x-ray.