What does it mean when my heart starts beating fast out of nowhere & I'm forced to gasp for breath?

Palpitations. This could be svt or just a sinus tachycardia. This should be evaluated.
Depends. In a young person, fast heart beat assoc. W/ shortness of breath could indicate an arrhythmia. In an older person, they could be due to an arrhythmia (irregular heart beat) or even coronary artery disease. Sometimes these symptoms can be seen in panic attacks, severe anemia, asthma, drugs, blood clot in the lungs, cardiomyopathy, or hyperthyroidism. Best to see doc to have further investigation.