Can symptoms of pulmonary embolism (chest tightness, heart palpitations) come and go, or will they be constant if it's a dangerous pe clot?

Yes. Many causes however, so exam and testing is required to be definitive. Small recurrent emboli are known and can be intermittent and variable, and eventually lethal.
variable. No predictable rules with pulmonary embolism. Sometimes the only symptom is a fast heart rate. If you are on birth control pills and having chest tightness + palpatations, you should consdier a pe and have the proper testing.