Does the second phase of menstrual cycle need to be at least 10 days to sustain a pregnancy?

Usually yes: The second phase of the menstrual cycle or the luteal phase usually needs to be at least twelve days long in order to maintain the pregnancy. After ovulation and fertilization the embryo stays within the fallopian tub for approximately 7 to 9 days before entering into the uterine cavity for implantation. If the uterine lining starts to break down prior to that then implantation will be unsuccessful.