Is it permissible to mortgage a house in order to acquire it, and what are the other legitimate (Shariah-compliant) ways to obtain a house in light of rising prices and the absence of Islamic banks?

There is no objection to a mortgage on its price with the selling party. The Islamic Fiqh Academy has deemed it permissible, and Imam Ahmad also adopted this view, differing from Ibn Hamid and Al-Shafi'i. Among the Sharia-compliant alternatives to usurious loans are: Istisna' (manufacturing contract), Murabaha (cost-plus financing), and Ijarah Muntahia Bil Tamleek (leasing ending in ownership).