May one Daven for a gentile or Mumar, such as for him to recover from an illness, or for him to have blessing in his Parnasa, or to find a spouse and have children?
It is permitted to Daven on behalf of a gentile or Mumar that does good acts for the Jewish people. This applies even if the gentile is an idolater. One is not to Daven on behalf of a gentile or Mumar that performs evil acts towards the Jewish people. [If the gentile does not perform either good or evil for the Jewish people, it is permitted to Daven on his behalf for the purpose of Darkei Shalom or other potential benefit. The Rebbe was publicly witnessed by Dollars to bless many gentiles of all walks of life. It however requires further analysis if one may even privately pray on behalf of a gentile when there is no benefit involved, and one is doing so simply out of his good will.]
Ger: A convert’s prayer on behalf of his parents does not help [to the same extent as a regular son who Daven’s for his father]. [However he may Daven for him if he so chooses, and such a prayer does have some affect.]
 Sefer Chassidim 790; Chaim Bayad [Falagi] 33; Darkei Teshuvah 151/29
 So is implied from Sefer Chassidim which includes a Jewish Mumar.
 So is implied from Sefer Chassidim ibid which only states not to Daven for one who does evil to the Jewish people. Vetzaruch Iyun as to why Davening for a gentile does not transgress “Lo Sechanem” [Y.D. 151/11]
 However this
 Yechaveh Daas 6/60
 Yechaveh Daas ibid