Sometimes start to cough because of a cold randomly and it get almost so bad I almost puke. I have asthma. Is this just a cough or a mini asthma attack?

Likely an attack. Viral upper respiratory tract infection is a very common trigger for asthma. This can be quite serious if you do not have an effective plan in place. After informing your physician, s/he will want to review your asthma action plan and controller medications. If this is occurring now, you may try the on-call service, but otherwise go to urgent care to check respiratory status and get treated.