Okey so my opponent sacrifices his Child of Alara with Gift of Immortality attached to it. I have made a copy of his Child of Alara in the previous turn with Progenitor Mimic.
Can my copy of Child of Alara destroy his Child of Alara revived with Gift of Immortality, or does my copies mass destruction trigger before his Child of Alara gets revived?
If the Child of Alara your opponent controls dies, both its triggered ability and that of Gift of Immortality will trigger (C.R. 603.2, 603.3).
If your opponent orders those abilities by placing the Gift of Immortality ability first (C.R. 603.3b, 112.8), and the Child of Alara ability then resolves, all nonland permanents will be destroyed (including the Child of Alara you control). This will in turn make Progenitor-Mimic-turned-Child-of-Alara's triggered ability trigger (which will go on the stack above the Gift of Immortality ability -- which hasn't resolved yet [review C.R. 116.4]). Then, once that ability resolves, again all nonland permanents will be destroyed. Note that all this doesn't affect the Gift of Immortality ability, which can still get to resolve.
Can my copy of Child of Alara destroy his Child of Alara revived with Gift of Immortality, or does my copies mass destruction trigger before his Child of Alara gets revived?
If your opponent orders those abilities by placing the Gift of Immortality ability first (C.R. 603.3b, 112.8), and the Child of Alara ability then resolves, all nonland permanents will be destroyed (including the Child of Alara you control). This will in turn make Progenitor-Mimic-turned-Child-of-Alara's triggered ability trigger (which will go on the stack above the Gift of Immortality ability -- which hasn't resolved yet [review C.R. 116.4]). Then, once that ability resolves, again all nonland permanents will be destroyed. Note that all this doesn't affect the Gift of Immortality ability, which can still get to resolve.