Those whose father is alive should NOT shave.
There isn't anything like this. You are not allowed to have a clean shave (shaving moustache) if your elder brother is alive. Only brahmins are not allowed to have a clean head, as they need shikha. others can have clean head without any issue.
We should not Shave on Thursday(because wealth might go away) and Saturday.
Its belief that you should not do activities like washing clothes, shaving, using soaps etc on days when someone in your family fasts on that day or has fasted previously. Thursday is common in this context as usually people fast on this day.
Basically as a sacrifice.
There are people who think that cutting hair on Saturdays help cut down Saade Sati of Shani Dev.
No, it doesn't. Actually people don't cut hair on saturdays. Neither do people purchase new things on saturday.
It is also believed that one should not wash their hair or clothes on a Thursday.
considered a sacrifice
There is also a myth for not cutting hair or shaving on Tuesday
Same because its considered a sacrifice.
Moreover anyone calling practices in our religion as myth or superstition is due to the western class of thinking. They made us look like idiots so that we underestimate ourselves.
Please note that its scientifically well proven by our archaeologist's & scientist's that we are much older civilizations than them, our thought processes and everything is much more advanced than them and hence our beliefs.
Traditions which have been here since millions of years (yes, Ramayana happened 1.7 million yr ago) cannot be defied by some western retards, and we must follow rules and regulations our forefathers have set for us, to have a healthy and prosperous life.