Today the Church begins her Chair of Unity Octave which she prays for all those who are not Catholic for their conversion to the true Catholic Faith. We pray the following prayers from the Feast of the Chair of St. Peter at Rome (Jan. 18) to the Feast of the Conversion of St. Paul (Jan. 25):
Official Prayer
Antiphon: That they all may be one, as Thou, Father, in Me and I in Thee; that they also may be one in Us; that the world may believe that Thou hast sent Me. (John 17:21)
V. I say unto thee that thou art Peter, R. And upon this Rock I will build My Church.
Let us pray: O Lord Jesus Christ, Who didst say to Thine Apostles: peace I leave with you, My peace I give unto you; look not upon my sins, but upon the faith of Thy Church, and vouchsafe unto her that peace and unity which are agreeable to Thy Will. Who livest and reignest God forever and ever. Amen.
An indulgence of 300 days during the octave of prayers for the unity of the Church from the Feast of the Chair of St. Peter in Rome to the Feast of the Conversion of St. Paul. A plenary indulgence on the usual conditions at the end of the devout exercise.
Other Prayers
Almighty and everlasting God, whose will it is that all men should be saved and that none should perish, look upon the souls that are deceived by the guile of Satan, in order that the hearts of them that have gone astray may put aside all the perverseness of heresy, and, being truly repentant, may return to the unity of Thy truth. Through Christ our Lord. Amen.
From the Roman Missal. An indulgence of 3 years.
Lord Jesus, merciful Saviour of the world, we humbly beseech Thee by Thy Most Sacred Heart, that all the straying sheep may turn unto Thee, the Shepherd and Bishop of their souls: Who livest and reignest world without end. Amen.
An indulgence of 500 days (Pope St. Pius X, rescript in his own hand, Oct. 26, 1905, S. P. Ap., June 5, 1933.
Speray's Catholicism in a Nutshell 
Do you know if the prayers in the below link are actually what the Church used before the eclipse? There is language used in them that makes me a bit hesitant, i.e. “separated brethren”, and “may be fully converted to Thee”.
Click to access chair-of-unity-8.pdf
I could only find a few of the prayers from the 1957 Raccolta. The prayer that uses “separated brethren” was approved in 1868. That phrase is also used in prayers in 1869 approved by Pope Pius IX.
I could not find the prayer that uses “May be fully converted to Thee.” However, I don’t think it’s problematic either since there is a partial conversion when the unbaptized join Protestant and Eastern Orthodox churches. They need to fully convert by becoming Catholic. It doesn’t mean that a partial conversion is good enough to be saved, but only that it’s better and closer to Catholicism than before. At least, that’s how I see it.
I could not find the source for all the prayers.
Is there indulgence for kissing a hand of a priest?
I don’t know, but I suspect there is an indulgence for it.