If someone gets a heart transplant, is it true that they can develop the old heart-owner's characteristics?

No: No, but it makes for a good movie.

No but:: Certain food allergies have been transferred from donor to recipient (e.g. Peanut allergy). This is thought to be due to cells responsible for the allergic reaction (mast cells) being transplanted as passenger cells along with the transplanted organ. But on the whole, no.

No: This is an urban legend or old tale. Solid organs can transmit diseases such as viral infections or cancers, but not characteristics like hair color, voice patterns, memories, favorite tastes, etc.