Does the retention of urine in the urethral opening after ritual washing invalidate prayer, and is it real urine? And does God forgive past sins after sincere repentance and changing for the better, even with occasional relapses into a bad habit and subsequent regret?

What you see on the organ may be residual water from istinja' (washing after relieving oneself), and not urine. The default assumption is that urine has not exited unless you are certain of it, so do not pay attention to doubts. If you are certain that urine has exited, repeat istinja' and perform wudu' (ablution). Your repentance is accepted, God willing, for Allah forgives all sins, and one who repents is like one who has no sin. Repent whenever you sin until Allah grants you sincere repentance. Accompany righteous people, and adhere to dhikr (remembrance of Allah) and du'a' (supplication).