What should I do if I am unable to completely cleanse myself of feces, and it is time for prayer or shortly before Friday prayer? And is it permissible for me to pray Tarawih with the obligatory prayer in this situation if my prayer is valid with certain procedures?

You must ensure that the impurity is completely removed. If the discharge of impurity is continuous and does not stop (sals), then you should clean the area, put on a protective pad, and perform ablution (wudu'). Whatever comes out after that will not be harmful, even if it occurs during prayer. You may pray the obligatory prayer, supererogatory prayers (nawafil), and sunnah prayers with this ablution. You should do this before every obligatory prayer when its time enters.