Is it obligatory to wash the private parts and their surroundings once, continuously, until the impurity is removed, or is it obligatory to wash them three times, as per the Maliki school of thought?

It is obligatory to remove impurity with any number of washes until it is gone. It is recommended, according to the Malikis, to first remove any harm (impurity) from the private parts by washing them once before ghusl if the impurity is removed thereby. The jurists have stipulated that it is desirable to begin by washing the hands, then removing impurity from the body, and that one should intend, when washing the area, to remove the major ritual impurity (hadath) or to permit what is otherwise forbidden, so that the ablution (wudu') is not invalidated. If one does not intend this, then it is obligatory to wash the area again with the intention. The most complete way is to first wash the areas of impurity, then wash those areas with the intention of ghusl for major ritual impurity.