Do the cells in a transplanted organ get replaced by your own?

No they definitely won't. Cystic fibrosis is a good example, when you are suffering from it you can't make the right type of ion channel - a pore in the surface of the cells - which means that the mucus gets really sticky and you can't get it out of your lungs easily. If it gets really bad you can transplant new lungs from a healthy donor and this work fine because they have a healthy copy of the cystic fibrosis gene in their stem cells which are making the lung wall cells, so the cells in the transplant work fine.