1- Sa'i should take place after a valid tawaf, be it after the going-forth tawaf, or the arrival tawaf for a pilgrim who has not yet stood at Arafa.
However, if standing at Arafa intervenes between the arrival tawaf and Sa'i, then one should not go between Safa and Marwa until he performs the going-forth tawaf
If the pilgrim goes between Safa and Marwa after the arrival tawaf then it is not recommended to repeat it after the going-forth tawaf.
2- To begin from Safa. Jaber (G. p. H) said: "When the Prophet (P. B. U. H) reached Safa he read:
"Sofa and Marwa are of Allah's rites, I begin with what Allah (G. B. H) began"
So, he (P. B. U. H) began with Safa".
[Muslim, Vol. 2, The Book of Hajj, Chapter 19/ 147].
If one begins at Marwa, then the first lap does not count, since
the second lap must begin at Marwa and the third lap at Safa and so on.
3- Going between Safa and Marwa must be done seven times.
It will be invalid if one neglects any of the seven times.
Going from Safa to Marwa is one lap and returning from Marwa to Safa is another lap.
One must cover the entire distance between Safa and Marwa, so if one leaves even a fraction of a single step then his Sa'i is invalid.
One who is walking must put his foot against the mountain when
starting, or finishing, without leaving any space in-between.
This is true if one does not climb Safa and Marwa, but if one does then it is the ideal since climbing is a confirmed sunna.
4- It should be within the designated walking path.
It is invalid to walk outside it. (pa:359 )
5- One's intention should not be diverted from Sa'i to something else. Therefore, if one goes between Safa and Marwa with the intention of catching an enemy then his Sa'i is invalid.
Racing with others in Sa'i, like some ignorant people do, is wrong if it is not coupled with the intention of making Sa'i.
Purity from both types of Hadath and other conditions needed for the validity of prayer are not necessary conditions for going between Safa and Marwa, because the walking path is not part of the mosque.