Is it sufficient to rinse with water and rub the anal area to remove traces of feces after performing istinja (ritual cleansing), and is it necessary to constantly check if anything has exited the anus?

The area of Istinja (definition)' (cleaning after defecation/urination) must be washed with water until the trace of impurity is removed. A strong presumption of purity is sufficient. It is not obligatory to use tissues before water. If you are certain that something exited after Istinja', you must repeat Istinja'. Otherwise, the default is purity, and do not pay attention to whispers (waswas).