Is it permissible to continue with the procedures for a housing loan, the interest of which is paid by another party, knowing that ultimately I will have to pay a small portion of this interest, and that is after I have already moved into the house, and after performing Istikhara and consulting some sheikhs?

It is not permissible to borrow with interest (riba), whether you pay the interest yourself or someone else pays it on your behalf. There is no difference whether it is from a bank, an institution, or an individual, for all riba is unlawful except in cases of dire necessity. Riba is among the gravest of major sins and the most severe in transgression, and Allah has threatened those who consume it with war from Him and His Messenger. Therefore, it is obligatory to avoid it and seek out lawful alternatives, such as the Tawarruq transaction with Islamic banks.