When do "immortal" Devil Fruits respawn? There is a question regarding the respawn time of regular Devil Fruits (DF), but I am interested in the possibility of creating an immortal army using the same one-time DF, namely the Ope Ope no Mi and the Yomi Yomi no Mi.

The Ope Ope no Mi can perform the "Perennial Youth Operation" which kills the DF user and gives eternal youth (and so potentially eternal life) to the subject. The Yomi Yomi no Mi enables the soul of the user to return to his body after his death.

Does the Ope Ope no Mi respawn after the Perennial Youth Operation (the user's death) or after the death of the operated person? Does the Yomi Yomi no Mi respawn after the "First" death or after the soul leaves this world forever?

At least for the Ope Ope no Mi there is no body left to contain the fruit's powers and after Doflamingo wanted to become immortal by Rosinante eating the fruit and performing the operation, the powers can not be stored in the operated person's body.

Could one create an army of immortals using one of this DFs?
No, these fruits cannot create an immortal army, they are not designed with that in mind. Even after giving someone eternal youth, that is all he will get, he will just remain "young" forever. Anyone will still be able to kill the user in a fight, he may die to an accident etc. The same applies Yomi Yomi no Mi, you get another life but you are not immortal. Also, even if someone has the Yomi Yomi no mi, and his head gets cut there is no body to return, because he will die again immeditally. 

Ope Ope no Mi respawns after the users death (the one who eat the fruit) because the person who gained eternal life has no "connection" with the fruit itself.