Can you get a chect infection, coughing, but u don't have flu or a cold? or can the chect infection with coughing be a sign of covid 19?

Yes: You can get bacterial infection of the chest with coughing such as pnumococcal pneumonia and bronchitis caused by a bacteria. So, chest infection with coughing can be a sign of covid 19 but other type infections can also cause chest infections and coughing. In summary other infectious agent can also cause chest infections and cough.

Chest infection: Coughing might be the symptom of a cold, the flu or COVID 19. Your doctor can help to address this.