Is interval running and weight lifting safe for someone with a weakened heart and valve one year after surgery?

Surgeon OK. Consult with your surgeon and express your desire to incorporate different types of exercise.
Medical clearance. You will need medical clearance to start up running, and particularly any higher intensity training like intervals. The question you raise really depends on how healthy your heart is now. Ask your cardiologist what you are able and not able to perform. Good luck.