Is taking an interest-bearing loan to secure housing considered a necessity that permits it, and will I bear sin for it? How can one act to repay the remaining loan given the lack of sufficient funds, and is it obligatory to sell one's car or gold to repay the debt?

Perhaps the tribulations are due to the ill effects of sin, specifically borrowing with interest (riba), which is a major sin. You both must hasten to sincere repentance, increase your istighfar (seeking forgiveness), and draw closer to Allah, for good deeds erase bad ones. You are not obligated to sell the car or gold to repay the loan, but you should hasten to repay it whenever you are able, especially if that will cancel or reduce the usurious interest. It appears there was no necessity to borrow with interest, as alternatives were available.