What is the ruling on someone who performs the funeral prayer and then severs the intention and claims to have prayed, or claims to have prayed without intention to avoid embarrassment? And what is the ruling on performing prayer movements without intention due to excessive impurity, and then repeating the prayer individually?

Shame that leads one to do what displeases Allah is reprehensible. Shame before Allah must take precedence over shame before people. Mimicking the actions of prayer is not permissible; you could have remained standing without mimicking. You must seek Allah's forgiveness and repent for what occurred. If the excessive flatulence is a result of obsessive thoughts, then disregard it. Wudu is not invalidated except by certain conviction of its emission. You should strive to seek treatment for the anxiety that causes excessive flatulence.