Is it permissible to use a piece of cotton to block the urethral opening and prevent pre-seminal fluid (madhy) from exiting during prayer, and does that affect the validity of the prayer or fasting, or must one wait for the flow of madhy to cease even if it means missing congregational prayer?

If the discharge of pre-seminal fluid (madhy) does not take up the entire prayer time, then wait for it to cease, perform ablution (wudu'), and pray, even if you miss the congregational prayer. Pray individually if you do not find another congregation. As for blocking the urethral opening to prevent madhy, if it involves severe pressure, it is an exaggeration and an extreme measure (ghuluw wa tanattu') that contradicts the ease of the religion. If it involves placing something to prevent its spread, then it is praying with impurity, and purity is necessary for the validity of the prayer. Therefore, wait for the madhy to cease, then wash your penis, perform ablution, and pray. Do not undertake what Allah has not obligated you to do, and beware of obsessive whispers (waswas).