Why should women perform Acamanam? What is the proper way of doing it? What are the mantras to be chanted during Acamanam? Is there a rule for performing it? Can it be performed during menstrual period?

As per the general rule, after taking bath, wearing pure clothes, after cleansing hands and legs one has to do Acamanam, this is applicable even for women.
Marriage for women is considered like an upanayanam and hence women must perform Acamanam after their marriage.
Women need not sit like men while doing it and it is also not in custom to do so. Women can simply chant the name Acyuta, Ananta, Govinda, omitting pranavam, namaH and partake the tirtham thrice.