Is it permissible to propose to a girl with whom the questioner had a romantic relationship that involved some sins, along with questioning the suitability of this decision for the future?

Both parties are obligated to engage in sincere repentance and sever the relationship. If they desire to marry, Islamic law does not prohibit it. However, it is advisable to wait to ensure her repentance and uprightness, as Islamic law encourages choosing a righteous woman who protects her husband in herself and his wealth. If her good character becomes apparent, then he may propose to her and continue performing Istikhara so that Allah may choose what is most suitable.