There's a lot of ways to ask this question, so I'll just jump right into the scenario.
SCENARIO A: There is a
Kathari Screecher
in my graveyard. I activate its Unearth ability, and after it resolves, I cast
Cloudshift
targeting that same Kathari Screecher. What happens next?
First possibility: The replacement effect generated from the Unearth ability replaces Cloudshift exiling the Screecher with the Screecher exiling itself. As a result, Cloudshift can't find the new object that the Screecher becomes after being exiled, and fails to return it to the battlefield. The Screecher remains exiled.
Second possibility: The replacement effect generated from the Unearth ability replaces Cloudshift exiling the Screecher with the Screecher exiling itself. However, Cloudshift is still able to find the new object that the Screecher becomes after being exiled, so it returns it to the battlefield. Unearth's delayed trigger ("At the beginning of your end step, exile [this].") eventually triggers, but since it refers to the Screecher as a past object which no longer exists, nothing happens when it resolves. The Screecher remains on the battlefield (until some other evil effect befalls it).
Third possibility: The replacement effect generated from the Unearth ability doesn't apply. Cloudshift resolves, "blink"ing the Screecher. Unearth's delayed trigger ("At the beginning of your end step, exile [this].") eventually triggers, but since it refers to the Screecher as a past object which no longer exists, nothing happens when it resolves. The Screecher remains on the battlefield (until some other evil effect befalls it).
SCENARIO B: There is a
Kathari Screecher
in my graveyard. I activate its Unearth ability. It resolves, then I cast
Oblivion Ring
. After it resolves, I target the Screecher with Oblivion Ring's trigger. After that trigger resolves, I finally cast
Erase
targeting that Oblivion Ring. What happens next?
First possibility: The replacement effect generated from the Unearth ability replaces Oblivion Ring's trigger exiling the Screecher with the Screecher exiling itself. As a result, the Screecher is not "exiled with Oblivion Ring". After Erase resolves and exiles the Oblivion Ring, its second ability triggers, but since nothing is exiled with Oblivion Ring, nothing is returned. The Screecher remains exiled.
Second possibility: The replacement effect generated from the Unearth ability replaces Oblivion Ring's trigger exiling the Screecher with the Screecher exiling itself. However, the Screecher is still "exiled with Oblivion Ring". After Erase resolves and exiles Oblivion Ring, its second ability triggers, returning Screecher to the battlefield. Unearth's delayed trigger ("At the beginning of your end step, exile [this].") eventually triggers, but since it refers to the Screecher as a past object which no longer exists, nothing happens when it resolves. The Screecher remains on the battlefield (until some other evil effect befalls it).
Third possibility: The replacement effect generated from the Unearth ability doesn't apply. Oblivion Ring exiles Screecher as normal. Erase then exiles Oblivion Ring, and the Screecher is returned. Unearth's delayed trigger ("At the beginning of your end step, exile [this].") eventually triggers, but since it refers to the Screecher as a past object which no longer exists, nothing happens when it resolves. The Screecher remains on the battlefield (until some other evil effect befalls it).
I'm not (yet) a certified judge, but I do understand the rules and logic of Magic quite well.
That said, I'm still human and can make mistakes. If what I say conflicts with a more-certified source, I'm probably wrong.