Constant coughing for a week, throwing up, pressure on chest, and a ton of pain. (not cancer). has asthma and eczema. could asthma be at fault?

Yes: See your dr. You need inhalers, probably 2 kinds. You may need an antibiotic. You may need a short course of Prednisone such as 20mg alternate days for 3 doses.

Yes indeed: In fact, "cough variant asthma" is one of the common explanations for what you describe. Did your albuterol inhaler touch it? Another entity is pertussis -- it's around thanks to the anti-immunization movement and your own immunization may not be enough to handle the load you meet. Yet another is medication allergy.