Yes, they often do.
No it shouldn't effect your menstrual cycle as you are not on any medication. However, it is possible for women who closely associate with each other or live together, for their cycles to arrive around the same time.
I don't have an answer but I have the same question.
No, not every woman has a menstrual cycle. Most healthy adult cis-women will menstruate, however women who have reached menopause will no longer have a menstrual cycle, nor will women who have had a hysterectomy - basically women who are no longer fertile. Women who use hormonal birth control don't have a menstrual cycle, as the purpose of hormonal birth control is to suppress the menstrual cycle so they don't ovulate - the bleeding women get is a withdrawal bleed, it mimics menstruation but isn't the same as menstruation. Many women were born biologically male, thus regardless of changes to their gender or genital reassignment surgery to change their sex, they are not capable of ovulation and thus don't have a menstrual cycle - although interestingly hormones given to many transsexual women can create symptoms similar to that of the menstrual cycle.
Yess
Women who spend a lot of time together sometimes end up having their menstrual cycles in sync. It simply means that you two are spending a lot of (hopefully happy) time together.
The menstrual cycle is the entire reproductive cycle, I think that you mean menstruation/period. Gymnasts do the same as every other girl, just use a pad, tampon, or menstrual cup.
The typical menstrual cycle length is around 28 days long, but everyone is different. Menstrual cycles will not always be the same length, up to a weeks variation from your average cycle is normal.
No
Yes, being sick can affect your menstrual cycle by making your period late. Stress, whether good or bad, can have the same effect.
Same x
Your menstrual cycle is the entire reproductive cycle including ovulation and menstruation - I think you mean menstruation of five days, not menstrual cycle. Menstruation will vary from one cycle to the next, the explanation is no more complex then that your body is not a machine so it will never have the exact same cycle every single time.
When a girl menstruates isn't determined by the calendar date, it is determines by her menstrual cycle - how long it takes for her body to go through the various reproductive phases. A typical menstrual cycle is 28 days so during this cycle it would be normal to menstruate roughly the same time every month, but women's cycles can vary by up to 7 days from one to the next - also different women have different cycle lengths.