1) When visiting a grave, some have the custom to say "Yehi Ratzon Sh'tehey Menuchaso Shel Ploni Po B'Kavod U'Zechuso Ya'amod Li, May it be the will [of Hashem] that the resting of the person here be with dignity and may his/her merit be beneficial for me." (Be'er Heitev Orach Chaim Siman 224:8)
The custom that people have to rip out some grass or pick up a stone and leave it on the Kever, is a show of respect to the one laying there, as it shows that people came to visit and pay their respect. (ibid.)
While there it is proper to place one's hand on the Kever. (ibid. The Arizal held that one should never go within 4 Amos of a grave, besides during the time of burial. See Mishna Berurah Siman 559:41. The Vilna Gaon, in the Igeres HaGra, wrote that people should not go to cemeteries at all, especially women. The prevalent custom, however, is that we do in fact go to cemeteries, and do in fact go within 4 Amos and even touch the Kevarim. However, women while they are a Niddah customarily do not go. See Mishna Berurah Siman 88:7 and Bais Baruch to Chayei Adam Siman 3:38)
2) The hand that should be placed on the Kever is the left hand. (See Kaf HaChaim Siman 224:42. The reason for this is kabalistic.)
There is no difference between left handed and right handed people regarding this, and both should only place their left hand on a Kever. (See Shu"t Be'er Moshe Vol. 2 Siman 5:11b)
It is important to remember when davening at a grave, especially that of a great Tzadik, to be extremely careful not to Chas V'Shalom pray "to" the Tzadik.
Rather, we Jews only pray to Hashem to answer us in the merit of the Tzadik, or, alternatively, we ask the Tzadik to go before the Kisei HaKavod and be a Meilitz Yosher (an advocate) for us and our families.
Ain Od M'Levado.