If I plan to get my medical tests done over vacation, will I need to go back for more for coughing up blood? I want to wrap this up in one week if possible.

You can't rush this. If you are coughing up blood, you may need anything from simple reassurance, to significant tests up to and including surgery, depending upon your age, risk factors (smoking), and the appearance of a chest x-ray. Sometimes addressing health concerns has to displace other priorities, because if they don't now, they most certainly will later on. I hope you are better soon.
Many causes. Many causes of spitting up blood, could be nose, lung, bronchi, tb, cancer, broken blood vessel from severe coughing.Could be serious so go and get it done.
Depends... The tests you will likely need include a cat scan of your chest and a bronchoscopy as well as pulmonary function testing. Further tests will depend on the results of these. It may not be possible to complete this work-up in a week but let your doctor know and I am sure he/she will try to accommodate you. Good luck!