Commercial banks (Stanbic, Zanaco and others) and the Zambia National Building Society (ZNBS). All are licensed by the Bank of Zambia.
Typically an employer introduction letter stating permanent or contract status, the latest 6 months of bank statements, NRC, and property documents or building plans for construction loans.
Yes. A balance transfer (refinancing) exists for exactly this, and ZNBS explicitly offers taking over mortgages from other institutions.
Construction loans pay out in stages against site inspections (ZNBS charges per visit); outright purchase pays the seller at once. Match the product to your situation.