Praise be to Allah, the Lord of the Worlds; and may His blessings and peace be upon our Prophet Muhammad and upon all his Family and Companions.
Banks have many ways to make profits without usury or interest. Allah made buying and selling permissible without setting a given price for selling.
Allah Says (interpretation of meaning): {…whereas Allâh has permitted trading and forbidden Ribâ (usury)….}[2:275].
One of these ways is Murabaha transection. I means as Ibn Qudamah said about this kind of transaction:
'It is the transection in which the capital and the profit are mentioned. The main condition of Murabaha transection is that the buyer and seller know the capital. The seller should say "my capital in it is such and such", or "I bought it with 100 and I sell to you with a profit of 10."
This method is permissible and there is no difference of opinion about its permissibility. Nobody said that it is disliked.
Therefore, it is permissible for a bank to buy the needs of its customers and to sell that to them by installments, or by setting a price that should be paid at a future date, at a profit that both sides of the contract agree on and set a dealine for it.
This profit should not be increased if the customer delays in paying.
It is permissible for the bank to take a mortgage from its customers to get its right if they delay paying back.
Your second question is already answered.
Allah knows best.