Is there a way in which a woman wouldn't have to wait 5 days after the start of menses, starting the count of 7 days earlier in order to go to mikva?... I would like to hear a clear answer for a Jewish woman according to scriptural and rabbinical laws and customs. How early can a woman go to mikvah after period?
It is important to make some clarification.
What are the "five days"?This requirement is based on the eventuality that the woman was clean (allowed to marital relationship) and has an intercourse immediately before the bleading. So, she is "Poletet Shichvat Zera", leaking male semen (this semen is still not putrefied, so theoretically able to fecund). The time needed to evacuate this semen do not exceed 72h, but the Minhag in many communities (as the OP mentionned) is to wait 5 days. After this time, the woman is allowed to count 7 days. _The explanation is that a semen leakage broke the count of 7 days and the woman need to count again from 1. The source of this Halacha is in Gemara, see here the Bet Yosef 196 paragraph 11 (יא) for a summary of Gemara.
How to skip the "five days"If a woman was already unclean (and so prohibited for marital relationship), e.g. if she do not go to Mikve after her precious menses. Now occurred a short menstruation of one day, It seems clear from poskim that, directly after the end of the bleeding, she is allowed to make hefsek Tahara and initiate the count. So the "five days" are skipped.
Sources in Halacha books.
See Shulchan Aruch 196, 11, and see in the next page the Pitche Teshuva sk 16. Some Poskim allow even she was clean before the bleeding, when her husband is not in the town (Bach, reported in Shach sk 20), but the Shach (op.cit.) rejected this opinion.
If the woman was not clean (allowed for marital relationship) before the occurrence of menses, and the menstruation was shorter than 5 days, when the bleeding stopped, she is immediately allowed to start to count the 7 days