Does the new heart mend to the old parts during heart transplant?

Somewhat: The removal includes atria and ventricles and leaves some atrial, aortic, pulmonary artery to match and sew to the new donor heart.

To a degree: Interesting question. The various body parts and organs are dependent on the blood flow or cardiac output from the heart. Depending upon the degree they were compromised with the old/damaged heart, they will be rejuvenated by the new heart. Most of the old heart is removed, so that there is little to 'mend.'.