Islam is built on ease towards people, Allah said:

“He did not make of religion a burden on you in any matter.” (Al-Hajj 22:78)

The biological way that made women able to be pregnant was meant to be accommodated in the Shari`ah of Islam. It will be stressful with the change of hormones, moods and emotions to keep up with such obligations like Salah and fasting. And that is why it was not obligated on females to fast or pray during their menstrual periods.

Let me correct you though on the Hajj topic. A female can do Hajj even with her monthly period. With medications you can take pills to regulate your period for that week or stay few days to finish certain things like tawaf. But during Hajj you can stand on `Arafah, sleep in Muzdalifah or Mina and throw pebbles even while you are on your period. Cleanliness from the menstrual period is not a requirement for all that.