What is the ruling on working in a non-Islamic bank in the تسهيلات (facilitation/credit) department given the difficult economic situation and the need to get married, and is emigrating to Sweden for work and marriage permissible under these circumstances?

Working in usurious banks is prohibited because it constitutes aiding in usury. It is not permissible to remain employed by them or to borrow from them except out of necessity, which means being on the brink of ruin and finding no other way to sustain oneself except through that which is forbidden, or being in a dire situation that can only be alleviated by that which is forbidden. If you cannot find a permissible job and wish to travel to non-Muslim countries, be aware that traveling there poses a risk to one's faith and morals. Working in a usurious bank—if no other option is available—is less harmful than going to countries where one's faith is feared to be at risk. The lesser of two evils is committed to avert the greater evil. However, if it is assumed that you can maintain your religious practices and do not fear temptation in non-Muslim countries, then it is permissible for you to travel there. Compare all these options and choose what is least harmful.