Who owns the Appraisal ?

Even though the borrower pays for the appraisal, the mortgage company owns it. This is because the mortgage company orders the appraisal on the borrower's behalf, and the appraiser lists that mortgage company on the appraisal report. However, the borrower has the right to receive a copy. It is at the mortgage company's discretion whether or not to give the borrower the original appraisal.

Can I use another mortgage company even after the appraisal has been completed ?

Yes. In most cases, changing your mortgage company does not mean you will have to pay for another appraisal. The first lender can transfer the appraisal to your new lender. Some appraisal firms may charge a small fee, however, because there is clerical work involved in editing the appraisal to reflect the new mortgage company. This fee is called an "Appraisal Retype Fee." The original mortgage company has the right to refuse to transfer the appraisal to another lender. In this event, you will need to get a new appraisal.