Is it permissible for me to obtain housing through an interest-based (usurious) loan to please my mother, who is pushing me to do so, knowing that my father expelled me from the house and mistreated me? Also, are the funds I save in my bank account from my salary considered permissible?

What the father is doing is not permissible شرعًا (Islamically) and is a squandering of Allah's وصية (command/bequest). Nevertheless, you must be patient and treat him well. Taking out a loan with interest (riba) is absolutely forbidden, but it is permissible in cases of dire necessity for essentials like housing, provided that you cannot find a owned or rented dwelling, and you do not have money to buy a dwelling, and you build only according to your need. As for depositing money in an interest-based (ribawi) bank, it is not permissible even without interest, because it constitutes cooperation in sin. The mother's statement that she will bear the sin is forbidden, because no one bears the sin of another.