I know each day was a "domain" of a god, and determined who your patron god was and what fundamental idea you were meant to express like knowledge, drunkeness, or chaos etc.
However I never find anything about the 5 bad days and why these days were cursed. What was the reasoning behind the cursed days and who were these terrible gods you would never want as your patron? Similarly, I cannot find anything about how a birthday changes your name.
What is stopping a parent from lying about the day of a child's birth with a fake birth name for example?
Can anyone shed light on this facet of life?
Ah ha, this is a question for me! I'm coming from a Kaqchikel Maya standpoint so I'll be using the Kaqchikel words for the calendars.
Disclaimer: I am not Mayan or Mesoamerican, I am white, but I have spent years studying this in my personal and academic life.
Mayan people did (and still do, in places) put a lot of stock into their birthday on the Chol Q'ij calendar. This calendar is what "Mayan astrology" is based off of. It works like two gears that turn together. One has 13 numbers on it and the other has 20 day names, so there are 260 combinations. Today (May 8) was 9 Ajpu but it's past sunset so tomorrow is 10 Imox, and at sunset tomorrow it'll be 11 Iq', up til 13 then it starts over again at 1, on and on again forever. It won't be 9 Ajpu on May 8 again for another 52 years. The day name is the most important part of the astrology here as they give the most personality and have the gods attached to them and such (in facts each day is thought of as a god or spirit in and of itself). There are also animal guardians (ruwäch q'ij) that become attached to a human soul the moment it's born. i can explain more about this if you want but that'd turn this comment into a novel.
The 5 bad days, the Wayeb', are the last days of the Chol 'Ab calendar, which is the 365 day civil calendar. It's a bit like if December 26-31 was bad luck, then January 1 had a feast. It's unlucky to be born on these days, but it doesn't cancel out any good luck you might have from your Chol Q'ij birthday. You're just probably going to have a harder time finding a midwife willing to come out to your home when you go into labor during Wayeb'.
Fun fact: my user name is my Mayan name, B'eleje' N'oj, which is from the Chol Q'ij calendar. It means Nine Woodpecker and basically means I'm (9) resilient and feminine, and (Woodpecker) obsessed with seeking and gaining knowledge, seek justice for people who are maligned or suffering, and very masculine. As a stubborn transmasculine activist and researcher... well... I've found some truths in the Chol Q'ij.